LOGARITHMIC GEOMETRY, MINIMAL FREE RESOLUTIONS AND TORIC ALGEBRAIC STACKS ISAMU IWANARI Abstract. In this paper we will introduce a certain type of morphisms of log schemes (in the sense of Fontaine, Illusie and Kato) and study their moduli. Then by applying this we define the notion of toric algebraic stacks, which may be regarded as torus emebeddings in the framework of algebraic stacks and prove some fundamental properties. Also, we study the stack-theoretic analogue of toroidal embeddings.

Introduction In this paper we will introduce a certain type of morphisms of log schemes (in the sense of Fontaine, Illusie, and Kato) and investigate their moduli. Then by applying this we define a notion of toric algebraic stacks over arbitrary schemes, which may be regarded as torus embeddings within the framework of algebraic stacks, and study some basic properties. Our notion of toric algebraic stacks gives a natural generalization of smooth torus embedding (toric varieties) over arbitrary schemes preserving the smoothness, and it is closely related to simplicial toric varieties. We first introduce a notion of the admissible and minimal free resolutions of a monoid. This notion plays a central role in this paper. This leads to define a certain type of morphisms of fine log schemes called admissible FR morphisms. (“FR” stands for free resolution.) We then study the moduli stack of admissible FR morphisms into a toroidal embedding endowed with the canonical log structure. One may think of these moduli as a sort of natural “stack-theoretic generalization” of the classical notion of toroidal embeddings. As promised above, the concepts of admissible FR morphisms and their moduli stacks yield the notion of toric algebraic stacks over arbitrary schemes. Actually in the presented work on toric algebraic stacks, admissible free resolutions of monoids and admissible FR morphisms play the role which is analogous to that of the monoids and monoid rings arising from cones in classical toric geometry. That is to say, the algebraic aspect of toric algebraic stacks is the algebra of admissible free resolutions of monoids. In a sense, our notion of toric algebraic stacks is a hybrid of the definition of toric varieties given in [4] and the moduli stack of admissible FR morphisms. Fix a base scheme S. Given a simplicial fan Σ with additional data called “level” n, we define a toric algebraic stack X(Σ,Σ0n ) . It turns out that this stack has fairly good properties. It is a smooth Artin stack of finite type over S with finite diagonal, whose coarse moduli space is the simplicial toric variety XΣ over Communicated by S. Mori. Received June 13, 2008. Revised February 2, 2009. 2000 Mathematics Subject Classification(s): Primary: 14M25; Secondary: 14A20. Key words: algebraic stacks, logarithmic geometry, toric geometry *Department of Mathematics, Faculty of Science, Kyoto University, Kyoto, 606-8502, Japan e-mail: [email protected]. 1

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S (see Theorem 4.6). Moreover it has a torus-embedding, a torus action functor and a natural coarse moduli map, which are defined in canonical fashions. The complement of the torus is a divisor with normal crossings relative to S. If Σ is non-singular and n is a canonical level, then X(Σ,Σ0n ) is the smooth toric variety XΣ over S. Thus we obtain the following diagram of (2)-categories, a bbbbbbbbb1

(Smooth toric varieties over S) \ b \\\\\\-

(Toric algebraic stacks over S) c

²

(Simplicial toric varieties over S) where a and b are fully faithful functors and c is an essentially surjective functor (see Remark 4.8). One remarkable point to notice is that working in the framework of algebraic stacks (including Artin stacks) allows one to have a generalization of smooth toric varieties over S that preserves the important features of smooth toric varieties such as the smoothness. There is another point to note. Unlike toric varieties, some properties of toric algebraic stacks depend very much on the choice of a base scheme. For example, the question of whether or not X(Σ,Σ0n ) is Deligne-Mumford depends on the base scheme. Thus it is natural to develop our theory over arbitrary schemes. Over the complex number field (and algebraically closed fields of characteristic zero), one can construct simplicial toric varieties as geometric quotients by means of homogeneous coordinate rings ([8]). In [5], by generalizing Cox’s construction, toric Deligne-Mumford stacks was introduced, whose theory comes from Cox’s viewpoint of toric varieties. On the other hand, roughly speaking, our construction stemmed from the usual definition of toric varieties given in, for example [18], [4], [10], [21], [6, Chapter IV, 2], in log-algebraic geometry, and it also yields a sort of “stacky toroidal embeddings”. We hope that toric algebraic stacks provide an ideal testing ground for problems and conjectures on stacks in many areas of mathematics, such as arithmetic geometry, algebraic geometry, mathematical physics, etc. This paper is organized as follows. In section 2, we define the notion of adimissible and minimal free resolution of monoids and admissible FR morphisms and investigate their properties. It is an “algebra” part of this paper. In section 3, we construct algebraic moduli stacks of admissible FR morphisms of toroidal embeddings with canonical log structures. In section 4, we define the notion of toric algebraic stacks and prove fundamental properties by applying section 2 and 3. Applications and further works. Let us mention applications and further works, which are not discussed in this paper. The presented paper plays a central role in the subsequent papers ([12], [13]). In [12], by using the results and machinery presented in this paper, we study the 2-category of toric algebraic stacks, and show that 2-category of toric algebraic stacks are equivalent to the category of stacky fans. Furthermore we prove that toric algebraic stacks defined in this paper have a quite nice geometric characterization in characteristic zero. In [13], we calculate the integral Chow ring of a toric algebraic stack and show that it is isomorphic to the Stanley-Reisner ring. As a possible application, we hope that our theory might be applied to smooth toroidal compactifications of spaces (including arithmetic schemes) that can not be smoothly compactified in classical toroidal geometry (cf. [4], [10], [20]). Notations And Conventions

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(1) We will denote by N the set of natural numbers, by which we mean the set of integers n ≥ 0, by Z the ring of rational integers, by Q the rational number field, and by R the real numbers. We write rk(L) for the rank of a free abelian group L. (2) By an algebraic stack we mean an algebraic stack in the sense of [19, 4.1]. All schemes, algebraic spaces, and algebraic stacks are assumed to be quasi-separated. We call an algebraic stack X which admits an ´etale surjective cover X → X , where X is a scheme a Deligne-Mumford stack. For details on algebraic stacks, we refer to [19]. Let us recall the definition of coarse moduli spaces and the fundamental existence theorem due to Keel and Mori ([17]). Let X be an algebraic stack. A coarse moduli map (or space) for X is a morphism π : X → X from X to an algebraic space X such that the following conditions hold. (a) If K is an algebraically closed field, then the map π induces a bijection between the set of isomorphism classes of objects in X (K) and X(K). (b) The map π is universal for maps from X to algebraic spaces. Let X be an algebraic stack of finite type over a locally noetherian scheme S with finite diagonal. Then a result of Keel and Mori says that there exists a coarse moduli space π : X → X with X of finite type and separated over S (See also [7] in which the Noetherian assumption is eliminated). Moreover π is proper, quasi-finite and surjective, and the natural map OX → π∗ OX is an isomorphism. If S 0 → S is a flat morphism, then X ×S S 0 → S 0 is also a coarse moduli map. Acknowledgement. I would like to thank Masao Aoki, Quo-qui Ito and Fumiharu Kato for their valuable comments. I also want to thank Tadao Oda for his interest and offering me many excellent articles on toric geometry. I would like to thank Institut de Math´ematiques de Jussieu for the hospitality during the stay where a certain part of this work was done. I am supported by Grant-in-Aid for JSPS Fellowships. 1. Toroidal and Logarithmic Geometry We first review some definitions and basic facts concerning toroidal geometry and logarithmic geometry in the sense of Fontaine, Illusie and K. Kato, and establish notation for them. We refer to [6, Chapter IV. 2] [10] [18] for details on toric and toroidal geometry, and refer to [15] [16] for details on logarithmic geometry. 1.1. Toric varieties over a scheme. Let N ∼ = Zd be a lattice and M := HomZ (N, Z) its dual. Let h•, •i : M × N → Z be the natural pairing. Let S be a scheme. Let σ ⊂ NR := N ⊗Z R be a strictly convex rational polyhedral cone and σ ∨ := {m ∈ MR := M ⊗Z R | hm, ui ≥ 0 for all u ∈ σ} its dual. (In this paper, all cones are assumed to be a strictly convex rational polyhedral, unless otherwise stated.) The affine toric variety (or affine torus embedding) Xσ associated to σ over S is defined by Xσ := Spec OS [σ ∨ ∩ M ] where OS [σ ∨ ∩ M ] is the monoid algebra of σ ∨ ∩ M over the scheme S. Let σ ⊂ NR be a cone. (We sometimes use the Q-vector space N ⊗Z Q instead of NR .) Let v1 , . . . , vm be a minimal set of generators σ. Each vi spans a ray, i.e., a 1-dimensional face of σ. The affine toric variety Xσ is smooth over S if and only if the first lattice points of R≥0 v1 , . . . , R≥0 vm form a part of basis of N (cf. [10, page 29]). In this case, we refer to σ as a nonsingular cone. The cone σ is simplicial if it is generated by dim(σ) lattice

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points, i.e., v1 , . . . , vm are linearly independent. Let σ be an r-dimensional simplicial cone in NR and v1 , . . . , vr the first lattice points of rays in σ. The multiplicity of σ, denoted by mult(σ), is defined to be the index [Nσ : Zv1 + · · · + Zvr ]. Here Nσ is the lattice generated by σ ∩ N . If the multiplicity of a simplicial cone σ is invertible on S, we say that the cone σ is tamely simplicial. If σ and τ are cones, we write σ ≺ τ (or τ  σ) to mean that σ is a face of τ . A fan (resp. simplicial fan, tamely simplicial fan) Σ in NR is a set of cones (resp. simplicial cones, tamely simplicial cones) in NR such that: (1) Each face of a cone in Σ is also a cone in Σ, (2) The intersection of two cones in Σ is a face of each. If Σ is a fan in NR , we denote by Σ(r) the set of r-dimensional cones in Σ, and denote by |Σ| the support of Σ in NR , i.e., the union of cones in Σ. (Note that the set Σ is not necessarily finite. Even in classical situations, infinite fans are important and they arises in various contexts such as constructions of degeneration of abelian varieties, the construction of hyperbolic Inoue surfaces, etc.) Let Σ be a fan in NR . There is a natural patching of affine toric varieties associated to cones in Σ, and the patching defines a scheme of finite type and separated over S. We denote by XΣ this scheme, and we refer it as the toric variety (or torus embedding) associated to Σ. A toric variety X contains a split algebraic torus T = Gnm,S = Spec OS [M ] as an open dense subset, and the action of T on itself extends to an action on XΣ . For a cone τ ∈ Σ we define its associated torus-invariant closed subscheme V (τ ) to be the union [ Spec OS [(σ ∨ ∩ M )/(σ ∨ ∩ τ0∨ ∩ M )] σÂτ

in XΣ , where σ runs through the cones which contains τ as a face and τ0∨ := {m ∈ τ ∨ |hm, ni > 0 for some n ∈ τ } (Affine schemes on the right hand naturally patch together, and the symbol /(σ ∨ ∩ τ0∨ ∩ M ) means the ideal generated by σ ∨ ∩ τ0∨ ∩ M ). We have Spec OS [(σ ∨ ∩ M )/(σ ∨ ∩ τ0∨ ∩ M )] = Spec OS [σ ∨ ∩ τ ⊥ ∩ M ] ⊂ Spec OS [σ ∨ ∩ M ] and the split torus Spec OS [τ ⊥ ∩ M ] is a dense open subset of Spec OS [σ ∨ ∩ τ ⊥ ∩ M ], where τ ⊥ = {m ∈ M ⊗Z R| hm, ni = 0 for any n ∈ τ }. (Notice that τ0∨ = τ ∨ − τ ⊥ .) For a ray ρ ∈ Σ(1) (resp. a cone σ ∈ Σ), we shall call V (ρ) (resp. V (σ)) the torus-invariant divisor (resp. torus-invariant cycle) associated to ρ (resp. σ). The complement XΣ − T set-theoretically equals the union ∪ρ∈Σ(1) V (ρ). Set Zτ = Spec OS [τ ⊥ ∩M ]. Then we have a natural stratification XΣ = tτ ∈Σ Zτ and for each cone τ ∈ Σ, the locally closed subscheme Zτ is a T -orbit. 1.2. Toroidal embeddings. Let X be a normal variety over a field k, i.e., a geometrically integral normal scheme of finite type and separated over k. Let U be a smooth Zariski open set of X. We say that a pair (X, U ) is a toroidal embedding (resp. good toroidal embedding, tame toroidal embedding) if for every closed point x in X there exist an ´etale neighborhood (W, x0 ) of x, an affine toric variety (resp. an affine simplicial toric variety, an affine tamely simplicial toric variety) Xσ over k, and an ´etale morphism f : W −→ Xσ .

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such that f −1 (Tσ ) = W ∩ U . Here Tσ is the algebraic torus in Xσ . 1.3. Logarithmic geometry. First of all, we shall recall some generalities on monoids. In this paper, all monoids will assume to be commutative with unit. Given a monoid P , we denote by P gp the Grothendieck group of P . If Q is a submonoid of P , we write P → P/Q for the cokernel in the category of monoids. Two elements p, p0 ∈ P have the same image in P/Q if and only if there exist q, q 0 ∈ Q such that p + q = p0 + q 0 . The cokernel P/Q has a monoid structure in the natural manner. A monoid P is finitely generated if there exists a surjective map Nr → P for some positive integer r. A monoid P is said to be sharp if whenever p + q = 0 for p, q ∈ P , then p = q = 0. We say that P is integral if the natural map P → P gp is injective. A finitely generated and integral monoid is said to be fine. An integral monoid P is saturated if for every p ∈ P gp such that np ∈ P for some n > 0, it follows that p ∈ P . An integral monoid P is said to be torsion free if P gp is a torsion free abelian group. We remark that a fine, saturated and sharp monoid is torsion free. Given a scheme X, a prelog structure on X is a sheaf of monoids M on the ´etale site of X together with a homomorphism of sheaves of monoids h : M → OX , where OX is viewed as a monoid under multiplication. A prelog structure is a log structure if the map ∗ ∗ h−1 (OX ) → OX is an isomorphism. We usually denote simply by M the log structure ∗ (M, h) and by M the sheaf M/OX . A morphism of prelog structures (M, h) → (M0 , h0 ) is a map φ : M → M0 of sheaves of monoids such that h0 ◦ φ = h. For a prelog structure (M, h) on X, we define its associated log structure (Ma , ha ) to be the push-out of ∗ h−1 (OX ) −−−→ M   y ∗ OX

in the category of sheaves of monoids on the ´etale site Xet . This gives the left adjoint functor of the natural inclusion functor (log structures on X) → (prelog structures on X). We say that a log structure M is fine if ´etale locally on X there exists a fine monoid and a map P → M from the constant sheaf associated to P such that P a → M is an isomorphism. A fine log scheme (X, M) is saturated if each stalk of M is a saturated monoid. We remark that if M is fine and saturated, then each stalk of M is fine and saturated. A morphism of log schemes (X, M) → (Y, N ) is a pair (f, h) of a morphism of underlying schemes f : X → Y and a morphism of log structures h : f ∗ N → M, where f ∗ N is the log structure associated to the composite f −1 N → f −1 OY → OX . A morphism (f, h) : (X, M) → (Y, N ) is said to be strict if h is an isomorphism. Let P be a fine monoid. Let S be a scheme. Set XP := Spec OS [P ]. The canonical log structure MP on XP is the fine log structure induced by the inclusion map P → OS [P ]. Let Σ be a fan in NR (N ∼ = Zd ) and XΣ the associated toric variety over S. Then we have an induced log structure MΣ on XΣ by gluing the log structures arising from the homomorphism σ ∨ ∩ M → OS [σ ∨ ∩ M ] for each cone σ ∈ Σ. Here M = HomZ (N, Z). We shall refer this log structure as the canonical log structure on XΣ . If S is a locally noetherian ∗ regular scheme, we have that MΣ = OXΣ ∩ i∗ OSpec OS [M ] where i : Spec OS [M ] → XΣ is the torus embedding (cf. [16, 11.6]).

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Let (X, U ) be a toroidal embedding over a field k and i : U → X be the natural immersion. Define a log structure αX : MX := OX ∩ i∗ OU∗ → OX on X. This log structure is fine and saturated and said to be the canonical log structure on (X, U ). 2. Free Resolution of Monoids 2.1. Minimal and admissible free resolution of a monoid. Definition 2.1. Let P be a monoid. The monoid P is said to be toric if P is a fine, saturated and torsion free monoid. Remark 2.2. If a monoid P is toric, there exists a strictly convex rational polyhedral cone σ ∈ HomZ (P gp , Z) ⊗Z Q such that σ ∨ ∩ P gp ∼ = P . Here the dual cone σ ∨ lies on gp P ⊗Z Q. Indeed, we see this as follows. There exists a sequence of canonical injective homomorphisms P → P gp → P gp ⊗Z Q. Define a cone C(P ) := {Σni=0 ai · pi | ai ∈ Q≥0 , pi ∈ P } ⊂ P gp ⊗Z Q. Note that it is a full-dimensional rational polyhedral cone (but not necessarily strictly convex), and P = C(P ) ∩ P gp since P is saturated. Thus the dual cone C(P )∨ ⊂ HomZ (P gp , Z) ⊗Z Q is a strictly convex rational polyhedral cone (cf. [10, (13) on page 14]). Hence our assertion follows. • Let P be a monoid and S a submonoid of P . We say that the submonoid S is close to the monoid P if for every element e in P , there exists a positive integer n such that n · e lies in S. • Let P be a toric sharp monoid, and let r be the rank of P gp . A toric sharp monoid P is said to be simplicially toric if there exists a submonoid Q of P generated by r elements such that Q is close to P . Lemma 2.3. (1) A toric sharp monoid P is simplicially toric if and only if we can choose a (strictly convex rational polyhedral) simplicial full-dimensional cone σ ⊂ HomZ (P gp , Z) ⊗Z Q such that σ ∨ ∩ P gp ∼ = P , where σ ∨ denotes the dual cone in P gp ⊗Z Q. (2) If P is a simplicially toric sharp monoid, then C(P ) := {Σni=0 ai · pi | ai ∈ Q≥0 , pi ∈ P } ⊂ P gp ⊗Z Q is a (strictly convex rational polyhedral) simplicial full-dimensional cone. Proof. We first prove (1). The “if” direction is clear. Indeed, if there exists such a simplicial full-dimensional cone σ, then the dual cone σ ∨ is also a simplicial full-dimensional cone. Let Q be the submonoid of P , which is generated by the first lattice points of rays on σ ∨ . Then Q is close to P . Next we shall show the “only if” part. Assume there exists a submonoid Q ⊂ P such that Q is close to P and generated by rk(P gp ) elements. By Remark 2.2 there exists a cone C(P ) = {Σni=0 ai · pi | ai ∈ Q≥0 , pi ∈ P } such that P = C(P ) ∩ P gp . Note that since P is sharp, C(P ) is strictly convex and full-dimensional. Thus σ := C(P )∨ ⊂ HomZ (P gp , Z) ⊗Z Q is a full-dimensional cone. It suffices to show that C(P ) is simplicial, i.e., the cardinality of the set of rays of C(P ) is equal to the rank of P gp . For any ray ρ of σ ∨ = C(P ), Q ∩ ρ is non-empty because Q is close to P . Thus Q can not be generated by any set of elements of Q whose cardinality is less than the cardinality of rays in σ ∨ .

LOG GEOMETRY AND TORIC ALGEBRAIC STACKS

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Thus we have rk(P gp ) ≥ #σ ∨ (1) (here we write #σ ∨ (1) for the cardinality of the set of rays of σ ∨ ). Hence σ ∨ = C(P ) is simplicial and thus σ is also simplicial. It follows (1). The assertion of (2) is clear. 2 Lemma 2.4. Let P be a toric sharp monoid. Let F be a monoid such that F ∼ = Nr for some r ∈ N. Let ι : P → F be an injective homomorphism such that ι(P ) is close to F . Then the rank of P gp is equal to the rank of F , i.e., rk(F gp ) = r. Proof. Note first that P gp → F gp is injective. Indeed, the natural homomorphisms P → F and F → F gp are injective. Thus if p1 , p2 ∈ P have the same image in F gp , then p1 = p2 . Hence P gp → F gp is injective. Since i(P ) is close to F , the cokernel of P gp → F gp is finite. Hence our claim follows. 2 Proposition 2.5. Let P be a simplicially toric sharp monoid. Then there exists an injective homomorphism of monoids i : P −→ F which has the following properties. (1) The monoid F is isomorphic to Nd for some d ∈ N, and the submonoid i(P ) is close to F . (2) If j : P → G is an injective homomorphism, and G is isomorphic to Nd for some d ∈ N, and j(P ) is close to G, then there exists a unique homomorphism φ : F → G such that the diagram i

/F ~ ~~ j ~~φ ~ ² Ä~

P

G

commutes. Furthermore if C(P ) := {Σni=0 ai · pi | ai ∈ Q≥0 , pi ∈ P } ⊂ P gp ⊗Z Q (it is a simplicial cone (cf . Lemma 2.3 (2)), then there exists a canonical injective map F → C(P ) that has the following properties: (a) The natural diagram P ²

i

/F z z zz zz |zz

C(P ) commutes, (b) Each irreducible element of F lies on a unique ray of C(P ) via F → C(P ). Proof. Let d be the rank of the torsion-free abelian group P gp . By Lemma 2.3, C(P ) is a full-dimensional simplicial cone in P gp ⊗Z Q. Let {ρ1 , . . . , ρd } be the set of rays in C(P ). Let us denote by vi the first lattice point on ρi in C(P ). Then for any element c ∈ C(P ) we have a unique representation of c such that c = Σ1≤i≤d ai · vi where ai ∈ Q≥0 for 1 ≤ i ≤ d. Consider the map qk : P gp ⊗Z Q → Q; p = Σ1≤i≤d ai · vi 7→ ak (ai ∈ Q for all i). Set Pi := qi (P gp ) ⊂ Q. It is a free abelian group generated by one element. Let

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pi ∈ Pi be the element such that pi > 0 and the absolute value of pi is the smallest in Pi . Let F be the monoid generated by p1 · v1 , . . . , pd · vd . Clearly, we have F ∼ = Nd and P ⊂ F ⊂ P gp ⊗Z Q. Note that there exists a positive integer bi such that pi = 1/bi for each 1 ≤ i ≤ d. Therefore bi · pi · vi = vi for all i, thus it follows that P is close to F . It remains to show that P ⊂ F satisfies the property (2). Let j : P → G be an injective homomorphism of monoids such that j(P ) is close to G. Notice that by Lemma 2.4, we have G ∼ = Nd . The monoid P has the natural injection ι : P → P gp ⊗Z Q. On the other hand, for any element e in G, there exists a positive integer n such that n · e is in j(P ). Therefore we have a unique injective homomorphism λ : G → P gp ⊗Z Q which extends P → P gp ⊗Z Q to G. Indeed, if g ∈ G and n ∈ Z≥1 such that n · g ∈ j(P ) = P , then we define λ(g) to be ι(n · g)/n (it is easy to see that λ(g) does not depend on the choice of n). The map λ defines a homomorphism of monoids. Indeed, if g1 , g2 ∈ G, then there exists a positive integer n such that both n · g1 and n · g2 lie in P , and it follows that λ(g1 + g2 ) = ι(n(g1 + g2 ))/n = ι(n · g1 )/n + ι(n · g2 )/n = λ(g1 ) + λ(g2 ). In addition, λ sends the unit element of G to the unit element of P gp ⊗Z Q. Since P gp ⊗Z Q ∼ = Qd , gp gp thus λ : G → P ⊗Z Q is a unique extension of the homomorphism ι : P → P ⊗Z Q. The injectiveness of λ follows from its definition. We claim that there exists a sequence of inclusions P ⊂ F ⊂ G ⊂ P gp ⊗Z Q. Since P is close to G and C(P ) is a full-dimensional simplicial cone, thus each irreducible element of G lies in a unique ray of C(P ). (For a ray ρ of C(P ), the first point of G ∩ ρ is an irreducible element of G.) On the other hand, we have Z≥0 · pi ⊂ qi (Ggp ) ∩ Q≥0 . This implies F ⊂ G. Thus we have (2). By the above construction, clearly there exists the natural homomorphism F → C(P ). The property (a) is clear. The property (b) follows from the above argument. Hence we complete the proof of our Proposition. 2 Definition 2.6. Let P be a simplicially toric sharp monoid. If an injective homomorphism of monoids i : P −→ F that satisfies the properties (1), (2) (resp. the property (1)) in Proposition 2.5, we say that i : P −→ F is a minimal free resolution (resp. admissible free resolution) of P . Remark 2.7. (1) By the observation in the proof of Proposition 2.5, if j : P → G is an admissible free resolution of a simplicially toric sharp monoid P , then there is a natural commutative diagram P ²

i

φ

/F / nn G n z n nn zz zznnnnnn z |zzvnnn

C(P )

such that φ ◦ i = j, where i : P → F is the minimal free resolution of P . Furthermore, all three maps into C(P ) are injective and each irreducible element of G lies on a unique ray of C(P ). (2) By Lemma 2.4, the rank of F is equal to the rank of P gp .

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(3) We define the multiplicity of P , denoted by mult(P ), to be the order of the cokernel of igp : P gp → F gp . If P is isomorphic to σ ∨ ∩ M where σ is a simplicial cone, it is easy to see that mult(P ) = mult(σ). Proposition 2.8. Let P be a simplicially toric sharp monoid and i : P → F ∼ = Nd its minimal free resolution. Consider the following diagram P

i

/ F ∼ Nd =

q

π

²

Q

j

²

/ Nr ,

where Q := Image(π ◦ i) and π : Nd → Nr is defined by (a1 , . . . , ad ) 7→ (aα(1) , . . . , aα(r) ). Here α(1), . . . , α(r) are positive integers such that 1 ≤ α(1) < · · · < α(r) ≤ d. Then Q is a simplicially toric sharp monoid and j is the minimal free resolution of Q. Proof. After reordering we assume that α(k) = k for 1 ≤ k ≤ r. First, we will show that Q is a simplicially toric sharp monoid. Since Q is close to Nr via j, Q is sharp. If ei denotes the i-th standard irreducible element, then for each i, there exists positive integers n1 , . . . , nr such that n1 · · · e1 , . . . , nr ·er ∈ Q and n1 · · · e1 , . . . , nr ·er generates a submonoid which is close to Q. Thus it suffices only to prove that Q is a toric monoid. Clearly, Q is a fine monoid. To see the saturatedness we first regard P gp and Qgp as subgroups of Qd = (Nd )gp ⊗Z Q and Qr = (Nr )gp ⊗Z Q respectively. It suffices to show Qgp ∩ Qr≥0 = Q. Since P is saturated, thus P = P gp ∩ Qd≥0 . Note that q gp : P gp → Qgp is surjective. It follows that P gp ∩ Qd≥0 → Qgp ∩ Qr≥0 is surjective. Indeed, let ξ ∈ Qgp ∩ Qr≥0 and ξ 0 ∈ P gp such that ξ = q gp (ξ 0 ). Put ξ 0 = (b1 , . . . , bd ) ∈ P gp ⊂ (Nd )gp = Zd . Note that bi ≥ 0 for 1 ≤ i ≤ r. Since P gp is a subgroup of Zd of a finite index, there exists an element ξ 00 = (0, . . . , 0, cr+1 , . . . , cd ) ∈ P gp such that ξ 0 + ξ 00 = (b1 , . . . , br , br+1 + cr+1 , . . . , bd + cd ) ∈ Zd≥0 . Then ξ = q gp (ξ 0 ) = q gp (ξ 0 + ξ 00 ). Thus P gp ∩ Qd≥0 → Qgp ∩ Qr≥0 is surjective. Hence Q is saturated. It remains to prove that Q ⊂ Nr is the minimal free resolution. It order to prove this, recall that the construction of minimal free resolution of P . With the same notation as in the first paragraph of the proof of Proposition 2.5, the monoid F is defined to be a free submonoid N·p1 ·v1 ⊕· · ·⊕N·pd ·vd of C(P ) where pk ·vk is the first point of C(P )∩ q˜k (P gp ) for 1 ≤ k ≤ d. Here the map q˜k : P gp ⊗Z Q → Q · vk is defined by Σ1≤i≤d ai · vi 7→ ak · vk (ai ∈ Q for all i). We shall refer this construction as the canonical construction. After reordering, we have the following diagram N · p 1 · v1 ⊕ · · · ⊕ N · p d · vd

/ C(P )

/ P gp ⊗Z Q

² / C(Q)

² / Qgp ⊗Z Q

π

²

N · p1 · v1 ⊕ · · · ⊕ N · pr · vr

where pk · vk is regarded as a point on a ray of C(Q) for 1 ≤ i ≤ r. Then pk · vk is the first point of C(Q) ∩ q˜k0 (Qgp ) for 1 ≤ k ≤ r, where q˜k0 : Qgp ⊗Z Q → Q · vk ; Σ1≤i≤r ai · vi 7→ ak · vk (note that for any c ∈ Qgp ⊗Z Q, there is a unique representation of c such that c = Σ1≤i≤r ai · vi where ai ∈ Q≥0 for 1 ≤ i ≤ r.). Then Q → Nr ∼ = N · p 1 · v1 ⊕ · · · ⊕ N · p r · vr is the canonical construction for Q, and thus it is the minimal free resolution. Hence we obtain our Proposition. 2

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Proposition 2.9. form

(1) Let ι : P → F be an admissible free resolution. Then ι has the i n ι=n◦i:P →F ∼ = Nd → Nd ∼ =F

where i is the minimal free resolution and n : Nd → Nd is defined by ei 7→ ni · ei . Here ei is the i-th standard irreducible element of Nd and ni ∈ Z≥1 for 1 ≤ i ≤ d. (2) Let σ be a full-dimensional simplicial cone in NR (N = Zd , M = HomZ (N, Z)) and σ ∨ ∩ M ,→ F the minimal free resolution (note that σ ∨ ∩ M is a simplicially toric sharp monoid). Then there is a natural inclusion σ ∨ ∩ M ⊂ F ⊂ σ ∨ . Each irreducible element of F lies on a unique ray of σ ∨ . This gives a bijective map between the set of irreducible elements of F and the set of rays of σ ∨ . Proof. We first show (1). By Remark 2.2 (1), there exist natural inclusions P ⊂ F ⊂ F ⊂ C(P ) where the first inclusion P ⊂ F is the minimal free resolution and the composite P ⊂ F ⊂ F is equal to ι : P → F . Moreover each irreducible element of the left F (resp. the right F ) lies on a unique ray of C(P ). Let {s1 , . . . , sd } (resp. {t1 , . . . , td }) denote images of irreducible elements of the left F (resp. the right F ) in C(P ). Since the rank of the free monoid F is equal to the cardinality of the set of rays of C(P ), thus there is a positive integer ni such that ni · ti ∈ {s1 , . . . , sd } for 1 ≤ i ≤ d. After reordering, we have ni · ti = si for each 1 ≤ i ≤ d. Therefore our assertion follows. To see (2), consider σ ∨ ∩ M ⊂ C(σ ∨ ∩ M ) ⊂ (σ ∨ ∩ M )gp ⊗Z Q = M ⊗Z Q ⊂ M ⊗Z R where C(σ ∨ ∩ M ) = {Σ1≤i≤m ai · si | ai ∈ Q≥0 , si ∈ σ ∨ ∩ M } is a simplicial full-dimensional cone by Lemma 2.3. The cone σ ∨ is the completion of C(σ ∨ ∩ M ) with respect to the usual topology on M ⊗Z R. Then the second assertion follows from Proposition 2.5 (b) and the fact that the rank of the free monoid F is equal to the cardinality of the set of rays of C(σ ∨ ∩ M ). 2 Let P be a toric monoid. Let I ⊂ P be an ideal, i.e., a subset such that P + I ⊂ I. We say that I is a prime ideal if P − I is a submonoid of P . Note that the empty set is a prime ideal. Set V = P gp ⊗Z Q. For a subset S ⊂ V , let C(S) be the (not necessarily strictly convex) cone define by C(S) := {Σ1≤i≤n ai · si | ai ∈ Q≥0 , si ∈ S}. To a prime ideal p ⊂ P we associate C(P − p). By an elementary observation, we see that C(P − p) is a face of C(P ) and it gives rise to a bijective correspondence between the set of prime ideals of P and the set of faces of C(P ) (cf. [25, Proposition 1.10]). Let P be a simplicially toric sharp monoid. Then the cone C(P ) ⊂ P gp ⊗Z Q is a strictly convex rational polyhedral simplicial full-dimensional cone (cf. Lemma 2.3). A prime ideal p ∈ P is called a height-one prime ideal if C(P − p) is a (dim P gp ⊗Z Q − 1)-dimensional face of C(P ), equivalently p is a minimal nonempty prime. In this case, for each height-one prime ideal p of P there exists a unique ray of C(P ), which does not lie in C(P − p). Let P → F be the minimal free resolution. Notice that the rank of F is equal to the cardinality of the set of rays of C(P ). Therefore taking account of Proposition 2.5 (b), there exists a natural bijective correspondence between the set of rays of C(P ) and the set of irreducible

LOG GEOMETRY AND TORIC ALGEBRAIC STACKS

11

elements of F . Therefore there exists the natural correspondences {The set of height-one prime ideals of P } ∼ =

²

{The set of rays of C(P )} ²

∼ =

{The set of irreducible elements of F }. Definition 2.10. Let P be a simplicially toric sharp monoid. Let I be the set of heightone prime ideals of P . Let j : P → F be an admissible free resolution of P . Let us denote by ei the irreducible element of F corresponding to i ∈ I. We say that j : P → F is an admissible free resolution of type {ni ∈ Z≥1 }i∈I if j is isomorphic to the composite i w P → F → F where i is the minimal free resolution and w : F → F is defined by ei 7→ ni ·ei . Note that admissible free resolutions of a simplicially toric sharp monoid are classified by their type. We use the following technical Lemma in the subsequent section. Lemma 2.11. Let P be a toric monoid and Q a saturated subomonoid that is close to P . Then the monoid P/Q (cf. section 1.3) is an abelian group, and the natural homomorphism P/Q → P gp /Qgp is an isomorphism. Proof. Clearly, P/Q is finite and thus it is an abelian group. We will prove that P/Q → P gp /Qgp is injective. It suffices to show that P ∩ Qgp = Q in P gp . Since P ∩ Qgp ⊃ Q, we will show P ∩ Qgp ⊂ Q. For any p ∈ P ∩ Qgp , there exists a positive integer n such that n · p ∈ Q because Q is close to P . Since Q is saturated, we have p ∈ Q. Hence P ∩ Qgp = Q. Next we will prove that P/Q → P gp /Qgp is surjective. Let p ∈ P gp . Take p1 , p2 ∈ P such that p = p1 − p2 in P gp . It is enough to show that there exists p0 ∈ P such that p0 + p2 ∈ Q. Since Q is close to P , thus our assertion is clear. Hence P/Q → P gp /Qgp is sujective. 2 2.2. MFR morphisms and admissible FR morphisms. The notions defined below play a pivotal role in our theory. Definition 2.12. Let (F, Φ) : (X, M) → (Y, N ) be a morphism of fine log-schemes. We say that (F, Φ) is an MFR (=Minimal Free Resolution) morphism if for any point x in X, the monoid F −1 N x¯ is simplicially toric and the homomorphism of monoids Φx¯ : F −1 N x¯ → Mx¯ is the minimal free resolution of F −1 N x¯ . Proposition 2.13. Let P be a simplicially toric sharp monoid. Let i : P → F be its minimal free resolution. Let R be a ring. Then the map i : P → F defines an MFR morphism of fine log schemes (f, h) : (Spec R[F ], MF ) → (Spec R[P ], MP ), where MF and MP are log structures induced by charts F → R[F ] and P → R[P ] respectively. Proof. Since MF and MP are Zariski log structures arising from F → R[F ] and P → R[P ] respectively, to prove our claim it suffices to consider only Zariski stalks of log structures i.e., to show that for any point of x ∈ Spec R[F ] the homomorphism h : f −1 MP,f (x) → MF,x is the minimal free resolution. Suppose that F = Nr ⊕ Nd−r and x ∈ Spec R[(Nd−r )gp ] ⊂ Spec R[Nd−r ] = Spec R[Nr ⊕Nd−r ]/(Nr −{0}) ⊂ Spec R[Nr ⊕Nd−r ]. Then f (x) lies in Spec R[P0gp ] ⊂ Spec R[P0 ] = Spec R[P ]/(P1 ) ⊂ Spec R[P ], where P0 is

12

ISAMU IWANARI pr1

the submonoid of elements whose images of u : P → F = Nr ⊕ Nd−r → Nr are zero and P1 is the ideal generated by elements of P , whose images of u are non-zero. Indeed, since x ∈ Spec R[Nr ⊕ Nd−r ]/(Nr − {0}), thus f (x) ∈ Spec R[P ]/(P1 ). For any p ∈ P0 , the image of i(p) in (Nd−r )gp is invertible, and thus f (x) ∈ Spec R[P0gp ]. Note that there exists the commutative diagram P

i

²

f −1 MP,f (x)

/ F = Nr ⊕ Nd−r

h

²

pr1

r /M F,x = N ,

where the vertical surjective homomorphisms are induced by the standard charts P → MP and F → MF respectively. Applying Proposition 2.8 to this diagram, it suffices to prove that f −1 MP,f (x) → MF,x is injective. Since there are a sequence of surjective gp gp maps P gp → P gp /P0gp → f −1 MP,f (x) and the inclusion f −1 MP,f (x) ⊂ f −1 MP,f (x) , thus it is enough to prove that for any p1 and p2 in P such that u(p1 ) = u(p2 ) the element p1 − p2 ∈ P gp lies in P0gp . To this aim, it suffices to show that({0} ⊕ (Nd−r )gp ) ∩ P gp ⊂ P0gp . Let C(P ) ⊂ P gp ⊗Z Q and C(P0 ) ⊂ P gp ⊗Z Q be cones spanned by P and P0 respectively. Then C(P ) ∩ P gp = P (cf. Remark 2.2), and the cone C(P0 ) is a face of C(P ). Indeed, identifying P gp ⊗Z Q with F gp ⊗Z Q, the cone C(P ) and C(P0 ) are generated by irreducible elements of F = Nd and {0} ⊕ Nd−r respectively. For any p ∈ C(P0 ) ∩ P gp there exists a positive integer n such that n · p lies in P0 . Taking account of the definition of P0 and C(P0 ) ∩ P gp ⊂ P , we have p ∈ P0 , and thus C(P0 ) ∩ P gp = P0 . Since C(P0 ) is a cone in P gp ⊗Z Q, we have P0gp ⊗Z Q ∩ P gp = P0gp . (We regard P0gp ⊗Z Q as a subspace of P gp ⊗Z Q.) This means that ({0} ⊕ (Nd−r )gp ⊗Z Q) ∩ P gp = P0gp . Thus we conclude that 2 ({0} ⊕ (Nd−r )gp ) ∩ P gp ⊂ P0gp . Hence we complete the proof. Lemma 2.14. Let R be a ring. Let X = Spec R[σ ∨ ∩M ] be the toric variety over R, where σ is a full-dimensional simplicial cone in NR (N = Zd ). Let MX denote the canonical log structure induced by σ ∨ ∩ M → R[σ ∨ ∩ M ] Let MX,¯x be the stalk at a geometric point x¯ → X, and let MX,¯x → F be the minimal free resolution. Then there exists a natural bijective map from the set of irreducible elements of F to the set of torus-invariant divisors on X on which x¯ lies. In particular, if σ ∨ ∩ M → H is the minimal free resolution, then there exists a natural bijective map from the set of irreducible elements of H to σ(1). Proof. Without loss of generality we may suppose that x¯ lies on the subscheme Spec R[σ ∨ ∩ M ]/(σ ∨ ∩ M ) ⊂ Spec R[σ ∨ ∩ M ]. Then we have MX,¯x = σ ∨ ∩ M . The set of torus-invariant divisors on which x¯ lies is {V (ρ)}ρ∈σ(1) , i.e., the set of rays of σ. For each ray ρ ∈ σ(1) the intersection ρ⊥ ∩ σ ∨ is a (dim σ − 1)-dimensional face. Since σ ∨ is simplicial, there is a unique ray of σ ∨ which does not lie in ρ⊥ ∩ σ ∨ . We denote this ray by ρ? . Then it gives rise to a bijective map σ(1) → σ ∨ (1); ρ 7→ ρ? . By Proposition 2.5, there is a natural embedding σ ∨ ∩ M ,→ F ,→ σ ∨ and each irreducible element of F lies on a unique ray of σ ∨ . It gives a bijective map from the set of irreducible elements of F to σ ∨ (1). Hence our assertion follows. 2

LOG GEOMETRY AND TORIC ALGEBRAIC STACKS

13

Definition 2.15. Let S be a scheme. Let N = Zd be a lattice and M := HomZ (N, Z) its dual. Let Σ be a fan in NR and XΣ the associated toric variety over S. Let n := {nρ }ρ∈Σ(1) be a set of positive integers indexed by Σ(1). A morphism of fine log schemes (f, φ) : (Y, M) → (XΣ , MΣ ) is called an admissible FR morphism of type n if for any geometric point y¯ → Y the homomorphism f −1 MP,f (¯y) → My¯ is isomorphic to the comi

n

posite f −1 MP,f (¯y) → F → F where i : f −1 MP,f (¯y) → F is the minimal free resolution and n : F → F is defined by eρ 7→ nρ · eρ . Here for a ray ρ ∈ Σ(1) such that f (¯ y ) ∈ V (ρ) we write eρ for the irreducible element of F corresponding to ρ (cf. Lemma 2.14). Proposition 2.16. Let P be a simplicially toric sharp monoid. Suppose that P = σ ∨ ∩ M where σ ⊂ NR is a full-dimensional simplicial cone. Let n := {nρ }ρ∈σ(1) be a set of positive integers indexed by σ(1). Let ι : P → F be an admissible free resolution defined n to be the composite P → F → F where P → F is the minimal free resolution and n : F → F is defined by eρ 7→ nρ · eρ for each ray ρ ∈ σ(1). Here eρ is the irreducible element of F corresponding to ρ (cf. Definition 2.10, Lemma 2.14). Let R be a ring. Let (f, φ) : (Spec R[F ], MF ) → (Spec R[P ], MP ) be the morphism induced by ι. Then (f, φ) is an admissible FR morphism of type n. Proof. Let us denote by (g, ψ) : (Spec R[F ], MF ) → (Spec R[F ], MF ) the morphism induced by n : F → F . Notice that for any geometric point x¯ → Spec R[F ] the canonical pr1 map F → MF,x and F → MF,g(¯x) are of the form F ∼ = Ns ⊕ Nt → Ns ∼ = MF,¯x and s t pr1 s ∼ ∼ F = N ⊕ N → N = MF,g(¯x) respectively for some s, t ∈ Z≥0 . Therefore ψ x¯ : Ns ∼ = −1 s ∼ g MF,g(¯x) → N = MF,¯x is the homomorphism induced by eρ 7→ nρ · eρ . Taking account of Proposition 2.13, our assertion follows from the definition of ι : P → F . 2 Proposition 2.17. Let R be a ring. Let P be a simplicially toric sharp monoid and ι : P → F = Nd an admissible free resolution of P . Let (f, h) : (S, M) → (XP := Spec R[P ], MP ) be a morphism of fine log schemes. Here MP is the canonical log structure on Spec R[P ] induced by P → R[P ]. Let c : P → MP be the standard chart. Let s¯ → S be a geometric point. Consider the following commutative diagram P c¯s¯

²

f −1 MP,¯s

ι

¯ s¯ h

/F ²

α

/M s¯

where c¯s¯ is the map induced by the standard chart such that α ´etale locally lifts to a chart. Then there exists an fppf neighborhood U of s¯ in which we have a chart ε : F → M such that the following diagram P c

²

f ∗ MP ε

ι

/F ε

h

²

/M

commutes and the composition F → M → Ms¯ is equal to α. If the order of the cokernel of P gp → F gp is invertible on R, then we can take an ´etale neighborhood U of s¯ with the above property.

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ISAMU IWANARI

Proof. Let γ := cs¯ : P → f ∗ MP,¯s be the chart induced by the standard chart P → MP . In order to show our assertion, we shall prove that there exists a homomorphism t : F → Ms¯, which is a lifting of α, such that t ◦ ι = hs¯ ◦ γ. To this aim, consider the following diagram 5 Ms¯

hkkkkk

f ∗ MP,¯s o

²

kkk

P

γ

² gp M s¯ 5 gp l l h lll l l l

o f ∗ Mgp P,¯ s

mm6 F

ι mmmm

m mmm

ξ

² ∼ gp F 7/ Z d 7 gp n ψ ooo n ι nn o oo ² nnnn∼ ooo a gp / d P Z φ

γ gp

where vertical arrows are natural inclusions and φ and ψ are isomorphisms chosen as follows. By elementary algebra, we can take the isomorphisms φ and ψ so that a := ψ ◦ ιgp ◦ φ−1 is represented by the (d × d)-matrix (aij ) with aii =: λi ∈ Z≥1 for 0 ≤ i ≤ d, and aij = 0 for i 6= j. Let us construct a homomorphism F gp → Mgp s¯ filling in the diagram. Let {ei }1≤i≤d be the canonical basis of Zd and put mi := hgp ◦ γ gp ◦ φ−1 (ei ). Let ni be an gp gp gp −1 element in Mgp s¯ such that q(ni ) = α (ψ (ei )) for 0 ≤ i ≤ d, where q : Ms¯ → Ms¯ is the natural projection. Then there exists a unit element ui in OS∗ such that ui + λi · ni = mi λi 0 d in Mgp s [T1 , . . . , Td ]/(Ti − ui )i=1 is a finite flat OS,¯ ss¯ for 0 ≤ i ≤ d. The algebra O := OS,¯ gp gp 0 algebra. If the order of cokernel of P → F is invertible on R, O is an ´etale OS,¯s -algebra. After the base change to O0 , there exists a homomorphism η : F gp → Mgp s¯ which is an gp gp gp gp gp extension of h ◦ γ . Since Ms¯ = Ms¯ ×Ms¯ Ms¯, the map t : F → Ms¯ ×Mgp Ms¯ = Ms¯ s ¯ defined by m 7→ (η(ξ(m)), α(m)) is a homomorphism which makes the diagram P

ι

/F

γ

²

f ∗ MP,¯s

hs¯

²

t

/ Ms¯,

commutative. Since P and F are finitely generated, this diagram extends to a chart in some fppf neighborhood of s¯. The last assertion immediately follows. 2 3. Moduli stack of admissible FR morphisms into a toroidal embedding 3.1. Moduli stack of admissible FR morphisms. Let (X, U ) be a toroidal embedding over a field k. Let us denote by I the set of irreducible components of X − U . Consider a triple (X, U, n), where n = {ni ∈ Z≥1 }i∈I . We shall refer to (X, U, n) as a toroidal embedding (X, U ) of level n. Let MX be the canonical log structure of (X, U ). The pair (X, MX ) is a fine saturated log scheme (cf. section 1.3). If we further assume that (X, U ) is a good toroidal embedding (cf. section 1.2), then for any point x on X, the stalk MX,¯x is a simplicially toric sharp monoid. Proposition 3.1. Let MX,¯x be the stalk at a geometric point x¯ → X. Let MX,¯x → F be the minimal free resolution. Then there exists a natural map from the set of irreducible elements of F to the set of irreducible components of X − U in which x¯ lies. Proof. Set P = MX,¯x . Note first that there exists the natural correspondence between irreducible elements of F and height-one prime ideals of P (see section 2). Therefore it

LOG GEOMETRY AND TORIC ALGEBRAIC STACKS

15

is enough to show that there exists a natural map from the set of height-one prime ideals of P to the set of irreducible components of X − U on which x¯ lies. Let {p1 , . . . , pn } be the set of height-one prime ideals of P . Let D = X − U be the reduced closed subscheme (each component is pure 1-codimensional) and ID be the ideal sheaf associated to D. If x denotes the image of x¯, and OX,x denotes its Zariski stalk, then the natural morphism Spec OX,¯x /ID OX,¯x → Spec OX,x /ID OX,x maps generic points to generic points. On the other hand, the log structure MX (cf. section 1) has a chart P → OX,¯x and the support of MX is equal to D, the underlying space of Spec OX,¯x /ID OX,¯x is naturally equal to Spec OX,¯x /(p1 OX,¯x ∩. . .∩pn OX,¯x ). (The closed subscheme Spec k[P ]/(p1 k[P ]∩. . .∩pn k[P ]) has the same underlying space as the complement Spec k[P ] − Spec k[P gp ].) Therefore it suffices to prove that for any height-one prime ideal pi , the closed subset Spec OX,¯x /pi OX,¯x is an irreducible component of Spec OX,¯x /ID OX,¯x . By [16, Theorem 3.2 (1)] there exists ∼ ˆX,¯x → ˆX,¯x can an isomorphism O k 0 [[P ]][[T1 , . . . , Tr ]] and the composite P → OX,¯x → O 0 0 be identified with the natural map P → k [[P ]][[T1 , . . . , Tr ]], where k is the residue field of OX,¯x . (Strictly speaking, [16] only treats the case of Zariski log structures, but the same proof can apply to the case of ´etale log structures.) Since P − pi is a submonoid ˆX,¯x /pi O ˆX,¯x is isomorphic to the integral domain and moreover it is a toric monoid, thus O ˆX,¯x is a flat OX,¯x -algebra and the natural map OX,¯x → k[[P − pi ]][[t1 , . . . , tr ]]. Note that O ˆX,¯x is injective, thus OX,¯x /pi OX,¯x → O ˆX,¯x /pi O ˆX,¯x is injective. Therefore OX,¯x /pi OX,¯x is O an integral domain. Hence Spec OX,¯x /pi OX,¯x is irreducible. Thus we obtain the natural map as desired. 2 Definition 3.2. Let (X, U, n = {ni ∈ Z≥1 }i∈I ) be a good toroidal embedding of level n over k. (We denote by I the set of irreducible components of X − U .) Let MX be the canonical log structure on X. An admissible FR morphism to (X, MX , n) (or (X, U, n)) is a morphism (f, φ) : (S, MS ) → (X, MX ) of fine log schemes such that for any geometric point s¯ → S the homomorphism φ : f −1 MX,f (¯s) → MS,¯s is isomorphic to ι

n

f −1 MX,f (¯s) → F → F, where ι is the minimal free resolution and n is defined by e 7→ ni(e) · e where e is an irreducible element of F and i(e) is an irreducible component of X − U to which e corresponds via Proposition 3.1. (We shall call such a resolution an admissible free resolution of type n at f (¯ s).) We define a category X(X,U,n) as follows. The objects are admissible FR morphisms to (X, MX , n). A morphism {(f, φ) : (S, M) → (X, MX )} → {(g, ψ) : (T, N ) → (X, MX )} in X(X,U,n) is a morphism of (X, MX )-log schemes (h, α) : (S, M) → (T, N ) such that α : h∗ N → M is an isomorphism. By simply forgetting log structures, we have a functor π(X,U,n) : X(X,U,n) → (X-schemes), which makes X(X,U,n) a fibered category over the category of X-schemes. This fibered category is a stack with respect to ´etale topology because it is fibered in groupoid and log structures are pairs of ´etale sheaves and their homomorphism on ´etale site. Furthermore according to the fppf descent theory for fine log structures ([22, Theorem A.1]), X(X,U,n) is a stack with respect to the fppf topology. Theorem 3.3. Let (X, U, n = {ni ∈ Z≥1 }i∈I ) be a good toroidal embedding of level n over a field k. Then the stack X(X,U,n) is a smooth algebraic stack of finite type over k with

16

ISAMU IWANARI

finite diagonal. The functor π(X,U,n) : X(X,U,n) → X is a coarse moduli map, which induces ∼ −1 an isomorphism π(X,U,n) (U ) → U . If we suppose that (X, U ) is tame (cf. section 1.2), and ni is prime to the characteristic of k for all i ∈ I, then the stack X(X,U,n) is a smooth Deligne-Mumford stack of finite type and separated over k. In the case of ni = 1 for all i ∈ I, we have the followings. ∼ −1 (1) The coarse moduli map induces an isomorphism π(X,U,n) (Xsm ) → Xsm when we denote by Xsm the smooth locus of X. (2) Suppose further that (X, U ) is tame. If there exists another functor f : X → X such that X is a smooth separated Deligne-Mumford stack and f is a coarse moduli map, then there exists a functor φ : X → X(X,U,n) such that the diagram φ

X HH / X(X,U,n) HH HH π(X,U,n) H f HHH ² # X commutes in the 2-categorical sense and such φ is unique up to a unique isomorphism. Remark 3.4. Moreover the stack X(X,U,n) has the following nice properties, which we will show later (because we need some preliminaries). (1) We will see that X(X,U,n) is a tame algebraic stack in the sense of [2] (see Corollary 3.10). (2) We will prove that the complement X(X,U,n) − U with reduced induced stack structure is a normal crossing divisor on X(X,U,n) (see section 3.5). 3.2. Before the proof of Theorem 3.3, we shall observe the case when (X, U ) is an affine simplicial toric variety Spec R[σ ∨ ∩M ] over a ring R, where σ is a full-dimensional simplicial cone in NR (N = Zd ). (For the application in section 4, we work over a general ring rather than a field.) Each ray ρ ∈ σ defines the torus-invariant divisor V (ρ). Consider the torus embedding X(σ, n) := (Spec R[σ ∨ ∩ M ], Spec R[M ], n) of level n = {nρ }ρ∈σ(1) . Set P := σ ∨ ∩ M (this is a simplicially toric sharp monoid). Let ι : P → F be the injective i n homomorphism defined to be the composite P → F → F , where i is the minimal free resolution and n : F → F is defined by eρ 7→ nρ · eρ , where eρ denotes the irreducible element of F corresponding to a ray ρ ∈ σ(1) (cf. Lemma 2.14). The cokernel F gp /ι(P )gp is a finite abelian group (and isomorphic to F/ι(P ) by Lemma 2.11). We view F gp /ι(P )gp as a finite commutative group scheme over R. We denote by G := (F gp /ι(P )gp )D its Cartier dual over R. We define an action m : Spec R[F ]×R G → Spec R[F ] as follows. Put π : F → F gp /ι(P )gp the canonical map. For each R-algebra A and A-valued point p : F → A (a map of monoids) of Spec R[F ], a A-valued point g : (F gp /ι(P )gp ) → A (a map of monoids) of G sends p to the map pg : F → A defined by the formula pg (f ) = p(f ) · g(π(f )) for f ∈ F . We denote by [Spec R[F ]/G] the stack-theoretic quotient associated to the groupoid m, pr1 : Spec R[F ] ×R G ⇒ Spec R[F ] (cf. [19, (10.13.1)]). By [7, section 3 and Theorem 3.1], the coarse moduli space for this quotient stack is Spec R[F ]G where R[F ]G = {a ∈ R[F ]| m∗ (a) = pr∗1 (a)} ⊂ R[F ]. Moreover the natural morphism R[P ] → R[F ]G is an isomorphism: Claim 3.4.1. The natural morphism R[P ] → R[F ]G is an isomorphism. In particular, the toric variety Spec R[P ] is a coarse moduli space for [Spec R[F ]/G].

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17

Proof. Note first that Γ(G) = R[F gp ]/(f − 1)f ∈P gp ⊂F gp is a finite free R-module, and m∗ : R[F ] → R[F ] ⊗R R[F gp ]/(f − 1)f ∈P gp ⊂F gp maps f ∈ F to f ⊗ f . Since pr∗1 maps f to f ⊗ 1, thus m∗ (Σf ∈F rf · f ) = pr∗1 (Σf ∈F rf · f ) (rf ∈ R) if and only if rf = 0 for f with f ∈ / P gp ⊂ F gp . Therefore it suffices to prove that P gp ∩ F = P . Clearly, we have P gp ∩ F ⊃ P , thus we will show P gp ∩ F ⊂ P . For any f ∈ P gp ∩ F , there exists a positive integer n such that n · f ∈ P . Since P is saturated, we have f ∈ P . Hence our claim follows. 2 Let MP be the canonical log structure on the toric variety Spec R[P ]. Proposition 3.5. There exists a natural isomorphism Φ : [Spec R[F ]/G] → XX(σ,n) over Spec R[P ]. The composite Spec R[F ] → [Spec R[F ]/G] → XX(σ,n) corresponds to the admissible FR morphism (χ, ²) : (Spec R[F ], MF ) → (Spec R[P ], MP ) which is induced by ι : P → F. Proof. Let [Spec R[F ]/G] → Spec R[P ] be the natural coarse moduli map. According to [22, Proposition 5.20 and Remark 5.21], the stack [Spec R[F ]/G] over Spec R[P ] is isomorphic to the stack S whose fiber over f : S → Spec R[P ] is the groupoid of triples (N , η, γ), where N is a fine log structure on S, η : f ∗ MP → N is a morphism of log structures, and γ : F → N is a morphism, which ´etale locally lifts to a chart, such that the diagram P c¯

²

f −1 MP

ι

/F

/

γ

²

N

commutes. Here we denote by c the standard chart P → MP . Notice that the last condition implies that η is an admissible FR morphism to (Spec R[P ], MP , n). Indeed, by Proposition 2.17 there exists fppf locally a chart γ 0 : F → N such that γ 0 induces γ, and γ 0 ◦ ι : P → F → N is equal to P → f ∗ MP → N . This means that there exists fppf locally on S a strict morphism (S, N ) → (Spec R[F ], MF ) such that the composite (S, N ) → (Spec R[F ], MF ) → (Spec R[P ], MP ) is equal to (f, η) : (S, N ) → (Spec R[P ], MP ). By Proposition 2.16 (Spec R[F ], MF ) → (Spec R[P ], MP ) is an admissible morphism to (Spec R[P ], MP , n), thus so is (f, η) : (S, N ) → (Spec R[P ], MP ). Therefore there exists a natural morphism Φ : [Spec R[F ]/G] → XX(σ,n) which forgets the additional data of the map γ : F → N . To show that Φ is essentially surjective, it suffices to see that every object in XX(σ,n) is fppf locally isomorphic to the image of Φ. Let (f, φ) : (S, N ) → (Spec R[P ], MP ) be an admissible FR morphism to (Spec R[P ], MP , n). By Proposition 2.16, for any geometric point s¯ → S the map f −1 MP,f (¯s) → N s¯ is isomorphic to g −1 MP,g(¯x) → MF,¯x , where g : Spec R[F ] → Spec R[P ] is induced by ι : P → F , and x¯ is a geometric point on Spec R[F ] which is lying over f (¯ s). Since r d−r pr1 r −1 ∼ F → MF,¯x has the form F = N ⊕ N → N , thus P → f MP,f (¯s) → N s¯ has the ι r d−r pr1 r ∼ form P → F = N ⊕ N → N , thus the essentially surjectiveness follows from Proposition 2.17. Finally, we will prove that Φ is fully faithful. To this end, it suffices to show that given two objects (h1 : f ∗ MP → N1 , γ1 : F → N 1 ) and (h2 : f ∗ MP → N2 , γ2 : F → N 2 ) in [Spec R[F ]/G](S), any morphism of log structures ξ : N1 → N2 , such that ξ ◦ h1 = h2 , has the property that ξ¯ ◦ γ1 = γ2 . It follows from the fact that P is close to F via ι and

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every stalk of N 2 is of the form Nr for some r ∈ N. Indeed for any f ∈ F , there exists a positive integer n such that n · f ∈ P . Since ξ ◦ h1 = h2 , we have ξ¯ ◦ γ1 (n · f ) = γ2 (n · f ). Since every stalk of N 2 is of the form Nr for some r ∈ N, we conclude that ξ¯◦γ1 (f ) = γ2 (f ). Finally, we will prove the last assertion. By Proposition 2.16, (χ, ²) is an admissible FR morphism of type n. By [22, Proposition 5.20], the projection Spec R[F ] → [Spec R[F ]/G] amounts to the triple (MF , ² : χ∗ MP → MF , F → MF ) over χ : Spec R[F ] → Spec R[P ]. Hence Φ((MF , ² : χ∗ MP → MF , F → MF )) = (MF , ² : χ∗ MP → MF ) and thus our claim follows. 2 Proposition 3.6. With the same notation as above, we have followings: (1) XX(σ,n) has finite diagonal. (2) The natural morphism XX(σ,n) → Spec R[P ] is a coarse moduli map. (3) XX(σ,n) is smooth over R. Proof. We first prove (1). The base change of the diagonal map [Spec R[F ]/G] → [Spec R[F ]/G] ×R [Spec R[F ]/G] by the natural morphism Spec R[F ] ×R Spec R[F ] → [Spec R[F ]/G]×R [Spec R[F ]/G] is isomorphic to Spec R[F ]×R G → Spec R[F ]×R Spec R[F ], which maps (x, g) to (x, xg ). Since pr1 : Spec R[F ] ×R G → Spec R[F ] is proper and pr1 : Spec R[F ] ×R Spec R[F ] → Spec R[F ] is separated, thus Spec R[F ] ×R G → Spec R[F ] ×R Spec R[F ] is proper. Clearly, it is also quasi-finite, so it is a finite morphism and we conclude that [Spec R[F ]/G] has finite diagonal. Hence by Proposition 3.5 XX(σ,n) has finite diagonal. The assertion (2) follows from Claim 3.4.1 and Proposition 3.5. Next we will prove (3). By Proposition 3.5, we have a finite flat cover Spec R[F ] → XX(σ,n) from a smooth R-scheme Spec R[F ], where F is isomorphic to Nr for some r ∈ N. Let V → XX(σ,n) be a smooth surjective morphism from a R-scheme V . Notice that the compr1 posite V ×XX(σ,n) Spec R[F ] → V → Spec R is smooth, and V ×XX(σ,n) Spec R[F ] → V is a finitely presented flat surjective morphism. Indeed the composite V ×XX(σ,n) Spec R[F ] → Spec R[F ] → Spec R is smooth. Thus by [9, IV Proposition 17.7.7], we see that V is smooth over R. Hence XX(σ,n) is smooth over R. 2 Remark 3.7. Let σ be a simplicial (not necessarily full-dimensional) cone in NR . Then there exists a splitting N ∼ = N 0 ⊕ N 00 such that σ ∼ = σ 0 ⊕ {0} ⊂ NR0 ⊕ NR00 where σ 0 is a full0 ∼ dimensional cone in NR . Thus Xσ = Xσ0 ×R Spec R[M 00 ], where M 00 = HomZ (N 00 , Z). The log structure Mσ is isomorphic to the pullback pr∗1 Mσ0 where pr1 : Xσ0 ×R Spec R[M 00 ] → Xσ . Therefore XX(σ,n) is an algebraic stack of finite type over R for arbitrary simplicial cone σ. Also, Proposition 3.6 holds for arbitrary simplicial cones. 3.3. Proof of Theorem 3.3. We will return to the proof of Theorem 3.3. Let R = k be a field. Proof of algebraicity. We will prove that X(X,U,n) is a smooth algebraic stack of finite type over a field k with finite diagonal. Clearly, one can assume that X(σ, n) := (X, U, n) = (Spec R[σ ∨ ∩ M ], Spec R[M ], n) (with the same notation and hypothesis as in section 3.2), where σ is a full-dimensional simplicial cone. Then by Proposition 3.5 and Proposition 3.6 (1) and (3), XX(σ,n) is a smooth algebraic stack of finite type over R with finite diagonal because XX(σ,n) is isomorphic to [Spec R[F ]/G]. Since the restriction MX |U is a trivial −1 log structure, thus we see that π(X,U,n) (U ) → U is an isomorphism. If we suppose that (X, U ) is tame and ni is prime to the characteristic of k for all i ∈ I, then the order of G = F gp /ι(P )gp is prime to the characteristic of k. Therefore the Cartier

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dual (F gp /ι(P )gp )D is a finite ´etale group scheme over k. Thus [Spec k[F ]/G] is a DeligneMumford stack over k (cf. [19, Remarque 10.13.2]). Therefore in this case X(X,U,n) is a Deligne-Mumford stack. Coarse moduli map for X(X,U,n) . Next we will prove that the natural map X(X,U,n) → X is a coarse moduli map for X(X,U,n) . By the above argument, we see that X(X,U,n) → X is a proper quasi-finite surjective morphism. Indeed, in the case when X is a toric variety, X(X,U,n) → X is a coarse moduli map by Proposition 3.6 (2). Moreover X(X,U,n) is integral and X(X,U,n) → X is generically an isomorphism. By [23, Corollary 2.9 (ii)], we conclude that X(X,U,n) → X is a coarse moduli map. 2 Assume that ni = 1 for all i ∈ I. Before the proof of (1) of Theorem 3.3, we prove the following Lemma. Lemma 3.8. Let (X, U ) be a toroidal embedding. Let MX be the canonical log structure of (X, U ). For every geometric point x¯ on Xsm , the monoid MX,¯x is free. Proof. Clearly, our claim is local with respect to ´etale topology. Hence, we may assume that for every closed point x of X, there exist an ´etale neighborhood W → X, a simplicially toric sharp monoid P , and a smooth morphism of w : W → Spec k[P ]. Note that MX |W is induced by the log structure on Spec k[P ] defined by P → k[P ]. Since X is smooth, thus after shrinking Spec k[P ] we may assume that Spec k[P ] is a smooth affine toric variety over k. Therefore Spec k[P ] has the form Spec k[Nr ⊕ Zs ] (cf. [10, p28]) and the log structure is induced by Nr → k[Nr ⊕ Zs ]. Hence our claim is clear. 2 ∼

−1 Proof of (1) in Theorem 3.3. To prove that π(X,U,n) (Xsm ) → Xsm , it suffices to show −1 that π(X,U,n) (Xsm ) is an algebraic spaces because π(X,U,n) is a coarse moduli map. Thus let us show that every MFR morphism (f, h) : (S, N ) → (Xsm , MX |Xsm ) has no nontrivial automorphism in X(X,U,n) . By Lemma 3.8 (1), each stalk of f −1 MX is a free monoid. Hence h : f ∗ MX → N is an isomorphism because (f, h) is an MFR morphism ¯ : f −1 MX → N is an isomorphism. Therefore it does not have a non-trivial and thus h −1 automorphism and thus the map π(X,U,n) (Xsm ) → Xsm is an isomorphism. 2

Proof of (2) in Theorem 3.3. Let p : Z → X be an ´etale cover by a separated scheme Z (note that Z ×X Z is also a separated scheme since X has finite diagonal and thus Z ×X Z → Z ×k Z is finite). Let Xsing be the singular locus of X and put V := Z − p−1 (f −1 (Xsing )). Note that since X is a normal variety, the codimension of Xsing is bigger than 1. Thus the codimension of p−1 (f −1 (Xsing )) is bigger than 1. Let pr1 , pr2 : Z ×X Z ⇒ Z be natural projections. By (1) in Theorem 3.3, f ◦ p |V : V → X and f ◦ p ◦ pri |V ×X V : V ×X V → X (i = 1, 2) are uniquely lifted to morphisms into X(X,U,n) . We abuse notation and denote by f ◦ p |V and f ◦ p ◦ pri |V ×X V (i = 1, 2) lifted morphisms into X(X,U,n) . Then by the purity lemma due to Abramovich and Vistoli ([1, 3.6.2] [3, Lemma 2.4.1]), f ◦ p |V and f ◦ p ◦ pri |V ×X V (i = 1, 2) are extended to Z and Z ×X Z respectively. These extensions are unique up to a unique isomorphism. Since f ◦ p ◦ pr1 = f ◦ p ◦ pr2 , there exists a unique morphism φ : X → X(X,U,n) such that π(X,U,n) ◦ φ ∼ 2 = f. 3.4. Stabilizer group schemes of points on X(X,U,n) . In this subsection, we calculate the stabilizer group schemes (i.e. automorphism group schemes) of points on the stack X(X,U,n) for a good toroidal embedding (X, U ) of level n = {ni }i∈I . For the definition of points on an algebraic stack, we refer to [19, Chapter 5]. In this subsection, by a geometric

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point to an algebraic stack X we mean a morphism Spec K → X with an algebraically closed field K. Let us calculate the stabilizer groups of points on X(X,U,n) . Let x¯ : Spec K → X(X,U,n) be a point on X(X,U,n) with an algebraically closed field K. Note that the point x¯ can be naturally regarded as the point on X via the coarse moduli map. Suppose that x¯ : Spec K → X(X,U,n) corresponds to an admissible FR morphism (π(X,U,n) ◦ x¯, h) : (Spec K, MK ) → (X, MX , n). Thus Isom(¯ x, x¯) is the group scheme over K, which represents the contravariant-functor G : (K-schemes) → (groups), {v : S → Spec K} 7→ {the group of isomorphisms of the log structure v ∗ MK which are compatible with v ∗ h : (π(X,U,n) ◦ x¯ ◦ v)∗ MX → v ∗ MK }. (We shall call Isom(¯ x, x¯) the ∗ stabilizer (or automorphism) group scheme of point x¯.) Set P = (π(X,U,n) ◦ x¯) MX and F = MK . Since K is algebraically closed, there exist isomorphisms (π(X,U,n) ◦ x¯)∗ MX ∼ = K∗ ⊕ P and MK ∼ = K ∗ ⊕F . Therefore we have G({S → Spec K}) = Homgroup (F gp /P gp , Γ(S, OS∗ )). (Note that by Lemma 2.11 the natural map F/P → F gp /P gp is an isomorphism.) Hence we conclude that G is the Cartier dual of F gp /P gp over K. Thus we have the following result: Proposition 3.9. Let x¯ : Spec K → X(X,U,n) be a point with an algebraically closed field K. Set y¯ = π(X,U,n) (¯ x). Then the stabilizer group scheme of x¯ is isomorphic to the Cartier gp gp dual of (F ) /MX,¯y over K, where MX,¯y → F is an admissible free resolution of type {ni } at y¯ (cf. Definition 3.2). Tame algebraic stacks. The recent paper [2] introduced the notion of tame algebraic stacks, which is a natural generalization of that of tame Deligne-Mumford stacks. We will show that the moduli stack X(X,U,n) is a tame algebraic stack. Let us recall the definition of tame algebraic stacks. Let X be an algebraic stack over a base scheme S. Suppose that the inertia stack X ×X ×S X X is finite over X . Let π : X → X be a coarse moduli map for X (Keel-Mori theorem implies the existence). Let QCoh X (resp. QCoh X) denote the abelian category of quasi-coherent sheaves on X (resp. X). The algebraic stack X is said to be tame if the functor π∗ : QCoh X → QCoh X is exact. By [2, Theorem 3.2], X is tame if and only if for any geometric point Spec K → X with an algebraically closed field K, its stabilizer group is a linearly reductive group scheme over K. By Proposition 3.9, for any geometric point Spec K → X(X,U,n) with an algebraically closed field K, its stabilizer group is diagonalizable, thus we obtain: Corollary 3.10. The algebraic stack X(X,U,n) is a tame algebraic stack. 3.5. Log structures on X(X,U,n) . Let (X, U, n = {ni }i∈I ) be a good toroidal embedding of level n and X(X,U,n) the associated stack. Let us define a canonical (tautological) log structure on X(X,U,n) . Let us denote by Lis-´et(X(X,U,n) ) the lisse-´etale site of X(X,U,n) . (Recall the definition of the lisse-´etale site. The underlying category of Lis-´et(X(X,U,n) ) is the full subcategory of X(X,U,n) -schemes whose objects are smooth X(X,U,n) -schemes. A collection of morphisms {fi : Si → S}i∈I of smooth X(X,U,n) -schemes is a covering family in Lis-´et(X(X,U,n) ) if the morphism ti fi : ti Si → S is ´etale surjective.) We define a log structure M(X,U,n) as follows. Let s : S → X(X,U,n) be a smooth X(X,U,n) scheme, i.e., an object in Lis-´et(X(X,U,n) ). This amounts exactly to an admissible FR morphism (f, φ) : (S, MS ) → (X, MX , n) such that f = π(X,U,n) ◦ s. By attaching to

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s : S → X(X,U,n) ∈ Lis-´et(X(X,U,n) ) the log structure MS , we define a fine and saturated log structure M(X,U,n) . We shall refer this log structure as the canonical log structure on X(X,U,n) . (For the notion of log structures on algebraic stacks, we refer to [22, section 5].) Moreover by considering the homomorphisms φ : f ∗ MX → MS , we have a natural morphism of log stacks (π(X,U,n) , Φ) : (X(X,U,n) , M(X,U,n) ) → (X, MX ). Proposition 3.11. Let (X, U, n) be a good toroidal embedding with a level n = {ni }i∈I , where I is the set of irreducible elements of X − U . Then we have the followings. (1) The subscheme D = X(X,U,n) − U with reduced closed subscheme structure is a normal crossing divisor. The log structure M(X,U,n) is isomorphic to the log structure arising from the divisor D = X(X,U,n) − U . (2) Suppose further that (X, U ) is a tame toroidal embedding (cf. section 1.2) and ni is prime to the characteristic of the base field for all i. The morphism (π(X,U,n) , Φ) : (X(X,U,n) , M(X,U,n) ) → (X, MX ) is a Kummer log ´etale morphism. We postpone the proof of this Proposition, and it will be given in section 4.3 because it follows from the case when (X, U ) is a toric variety. Remark 3.12. One may regard X(X,U,n) as a sort of “stacky toroidal embedding” endowed with the log structure M(X,U,n) . 4. Toric algebraic stacks We define toric algebraic stacks. 4.1. Some combinatorics. Let N = Zd be a lattice and M = HomZ (N, Z) the dual lattice. Let h•, •i : M × N → Z be the dual pairing. • A pair (Σ, n = {nρ }ρ∈Σ(1) ) is called a simplicial fan with a level structure n if Σ is simplicial fan in NR and n = {nρ }ρ∈Σ(1) is the set of positive integers indexed by the set of rays Σ(1). • A pair (Σ, Σ0 ) is called a stacky fan if Σ is simplicial fan in NR and Σ0 is a subset of |Σ| ∩ N such that for any cone σ in Σ the restriction σ ∩ Σ0 is a submonoid of σ ∩ N which has the following properties: (i) σ ∩ Σ0 is isomorphic to Nr where r = dim σ, (ii) σ ∩ Σ0 is close to σ ∩ N . Put another way. If (Σ, Σ0 ) is a stacky fan and ρ is a ray of Σ, then there exists the first point wρ of ρ ∩ Σ0 . Since σ ∈ Σ is simplicial and Ndim σ ∼ = σ ∩ Σ0 is 0 a submonoid that is close to σ ∩ N , thus σ ∩ Σ is the free monoid ⊕ρ∈σ(1) N · wρ (⊂ σ). (Each irreducible element of the monoid σ ∩ Σ0 lies on a unique ray of σ.) Therefore the data Σ0 is determined by the set of points {wρ }ρ∈Σ(1) . Let us denote by vρ the first lattice point on a ray ρ ∈ Σ(1). For a ray ρ ∈ Σ(1), if nρ · vρ (nρ ∈ Z≥0 ) is the first point wρ of Σ0 ∩ ρ, then we shall call nρ the level of Σ0 on ρ. • Let (Σ, n = {nρ }ρ∈Σ(1) ) be a simplicial fan with a level structure n = {nρ }. The free-net of Σ associated to level n is the subset Σ0n ⊂ |Σ| ∩ N such that for each cone σ ∈ Σ the set σ ∩ Σ0n is the free submonoid generated by {nρ · vρ }ρ∈σ(1) , where vρ denotes the first lattice point on a ray ρ ∈ Σ(1). (This free-net may be regarded as the geometric realization of the level n.) The canonical free-net Σ0can of Σ is the free-net associated to the level {nρ = 1}ρ∈Σ(1) . Note that for any stacky fan (Σ, Σ0 ) there exists a unique level n = {nρ }ρ∈Σ(1) such that (Σ, Σ0 ) = (Σ, Σ0n ).

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• Let S be a scheme (or a ringed space). A stacky fan (Σ, Σ0n ) is called tame over S if for any cone σ ∈ Σ the multiplicity mult(σ) is invertible on S, and for any ρ ∈ Σ(1) the level nρ is invertible on S. For a stacky cone (σ, σn0 ), we define the multiplicity, denoted by mult(σ, σn0 ), of (σ, σn0 ) to be mult(σ) · Πρ∈σ(1) nρ . A stacky fan (Σ, Σ0n ) is tame over S if and only if mult(σ, σn0 ) is invertible on S for any cone σ ∈ Σ. • A stacky fan (Σ, Σ0n ) is called complete if Σ is a finite and complete fan, i.e., Σ is a finite set and the support |Σ| is the whole space NR . Remark 4.1. The notion of stacky fans was first introduced in [5]. 4.2. Toric algebraic stacks. Fix a base scheme S. Definition 4.2. Let (Σ ⊂ NR , n = {nρ }ρ∈Σ(1) ) be a simplicial fan with a level structure n. Let (Σ, Σ0n ) be the associated stacky fan. Define a fibered category X(Σ,Σ0n ) −→ (S-schemes) as follows. The objects over a S-scheme X are triples (π : S → OX , α : M → OX , η : S → M) such that: (1) S is an ´etale sheaf of sub-monoids of the constant sheaf M on X determined by M = HomZ (N, Z) such that for every geometric point x¯ → X, Sx ∼ = Sx¯ . Here x ∈ X is the image of x¯ and Sx (resp. Sx¯ ) denotes the Zariski (resp. ´etale) stalk. (The condition Sx ∼ = Sx¯ for every x¯ means that the ´etale sheaf S is arising from the Zariski sheaf S|XZar .) (2) π : S → OX is a map of monoids where OX is a monoid under multiplication. (3) For s ∈ S, π(s) is invertible if and only if s is invertible. (4) For each point x ∈ X, there exists some (and a unique) σ ∈ Σ such that Sx = σ ∨ ∩M . (5) α : M → OX is a fine log structure on X. (6) η : S → M is a homomorphism of sheaves of monoids such that π = α◦η, and for each geometric point x¯ → X, the homomorphism η¯ : S x¯ = (S/(invertible elements))x¯ → Mx¯ is isomorphic to the composite r

t

S x¯ ,→ F ,→ F, where r is the minimal free resolution of S x¯ and t is defined by eρ 7→ nρ · eρ where eρ denotes the irreducible element of F corresponding to a ray ρ ∈ Σ(1) (see Lemma 4.3) and nρ is the level of Σ0 on ρ. A set of morphisms (π : S → OX , α : M → OX , η : S → M) → (π 0 : S 0 → OX , α0 : M0 → OX , η 0 : S 0 → M0 ) over X is the set of isomorphisms of log structures φ : M → M0 such that φ ◦ η = η 0 : S = S 0 → M0 if (S, π) = (S 0 , π 0 ) and is the empty set if (S, π) 6= (S 0 , π 0 ). With the natural notion of pullbacks, X(Σ,Σ0n ) is a fibered category. According to [4, Theorem on page 10], for any S-scheme X, there exists an isomorphism HomS-schemes (X, XΣ ) ∼ = { all pairs (S, π) on X satisfying (1), (2), (3), (4)}, which commutes with pullbacks. Here XΣ is the toric variety associated to Σ over S. Therefore there exists a natural functor π(Σ,Σ0n ) : X(Σ,Σ0n ) −→ XΣ

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23

which simply forgets the data α : M → OX and η : S → M. Moreover α : M → OX and η : S → M are morphisms of the ´etale sheaves and thus X(Σ,Σ0n ) is a stack with respect to the ´etale topology. ∗ ∗ ) determine a substack of ,→ OX , π : M → OX Objects of the form (π : M → OX , OX X(Σ,Σ0n ) , i.e., the natural inclusion i(Σ,Σ0n ) : TΣ = Spec OS [M ] ,→ X(Σ,Σ0n ) . We shall call i(Σ,Σ0n ) the canonical torus embedding. This commutes with the torus-embedding iΣ : TΣ ,→ XΣ . Lemma 4.3. With the same notation as in Definition 4.2, let e be an irreducible element in F and let n be a positive integer such that n · e ∈ r(S x¯ ). Let m ∈ Sx¯ be a lifting of n · e. Suppose that Sx¯ = σ ∨ ∩ M . Then there exists a unique ray ρ ∈ σ(1) such that hm, vρ i > 0. Here vρ is the first lattice point of ρ, and h•, •i is the dual pairing. It does not depend on the choice of liftings. Moreover this correspondence defines a natural injective map {Irreducible elements of F } → Σ(1). Proof. Since the kernel of Sx¯ → S x¯ is σ ⊥ ∩ M , thus hm, vρ i does not depend upon the choice of liftings m. Taking a splitting N ∼ = N 0 ⊕ N 00 such that σ ∼ = σ 0 ⊕ {0} ⊂ NR0 ⊕ NR00 where σ 0 is a full-dimensional cone in NR0 , we may and will assume that σ is a full-dimensional cone, i.e., σ ∨ ∩ M is sharp. By Proposition 2.9 (2), there is a natural embedding σ ∨ ∩ M ,→ F ,→ σ ∨ such that each irreducible element of F lies on a unique ray of σ ∨ . It gives rise to a bijective map from the set of irreducible elements of F to σ ∨ (1). Since σ and σ ∨ are simplicial, we have a natural bijective map σ ∨ (1) → σ(1); ρ 7→ ρ? , where ρ? is the unique ray which does not lie in ρ⊥ . Therefore the composite map from the set of irreducible elements of F to σ(1) is a bijective map. Hence it follows our claim. 2 Remark 4.4. (1) In what follows, we refer to a homomorphism π : S → OX with properties (1), (2), (3), (4) in Definition 4.2 as a skeleton. If π : S → OX corresponds to X → XΣ , then π : S → OX is called the skeleton for X → XΣ . (2) Let (Σ, Σ0can ) be a stacky fan such that Σ is non-singular and Σ0can is the canonical free net. Then X(Σ,Σ0can ) is the toric variety XΣ over S. Indeed, for any object (π : S → OX , α : M → OX , η : S → M) in X(Σ,Σ0can ) and any point x ∈ X, the monoid S x¯ has the form Nr for some r ∈ N. Thus in this case, (5) and (6) in Definition 4.2 are vacant. (3) Toric algebraic stack can be constructed over Z and pull back from there to any other scheme. Therefore, for the proof of Proposition 4.5 and Theorem 4.6 (1), (2), (3), we may assume that the base scheme is Spec Z. Torus Action functor. The torus action functor a : X(Σ,Σ0n ) × TΣ −→ X(Σ,Σ0n ) is defined as follows. Let ξ = (π : S → OX , α : M → OX , η : S → M) be an object in X(Σ,Σ0n ) . Let φ : M → OX be a map of monoids from a constant sheaf M on X to OX , i.e., an X-valued point of TΣ := Spec OS [M ]. Here OX is viewed as a sheaf of monoids under multiplication. We define a(ξ, φ) by (φ · π : S → OX , α : M → OX , φ · η : S → M), where φ · π(s) := φ(s) · π(s) and φ · η(s) := φ(s) · η(s). (Note that S is a subsheaf of the constant sheaf of M .) Let h : M1 → M2 be a morphism in X(Σ,Σ0n ) × TΣ from (ξ1 , φ) to (ξ2 , φ), where ξi = (π : S → OX , α : Mi → OX , ηi : S → Mi ) for i = 1, 2, and

24

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φ : M → OX is an X-valued point of TΣ . We define a(h) to be h. It gives rise to the functor a : X(Σ,Σ0n ) ×TΣ −→ X(Σ,Σ0n ) over (S-schemes), which makes the following diagrams IdX

(Σ,Σ0 )

×m

X(Σ,Σ0n ) × TΣ × TΣ −−−−−n−−→ X(Σ,Σ0n ) × TΣ   a×Id a T y y Σ X(Σ,Σ0n ) × TΣ

a

−−−→

X(Σ,Σ0n ) ,

a

X(Σ,Σ0n ) × TΣ −−−→ X(Σ,Σ0n ) x x IdX 0 IdX ×e 0   (Σ,Σn ) (Σ,Σn ) X(Σ,Σ0n )

X(Σ,Σ0n ) .

commutes in the strict sense, i.e., a◦(a×IdTΣ ) = a◦(IdX(Σ,Σ0 ) ×m) and a◦(e×IdX(Σ,Σ0 ) ) = n n IdX(Σ,Σ0 ) , where m : TΣ × TΣ → TΣ is the natural action and e : S → TΣ is the unit section. n Thus the functor a defines an action of TΣ on X(Σ,Σ0n ) which extends the action of TΣ on itself to the whole stack X(Σ,Σ0n ) Here the notion of group actions on stacks is taken in the sense of [24, Definition 1.3]. This action makes the coarse moduli map π(Σ,Σ0n ) : X(Σ,Σ0n ) → XΣ torus-equivariant. Proposition 4.5. Let (Σ, n = {nρ }ρ∈Σ(1) ) be a simplicial fan Σ with level n. Then there exists a canonical isomorphism between the stack X(Σ,Σ0n ) and the moduli stack X(XΣ ,TΣ ,n) of admissible FR morphisms to XΣ of type n (cf. Definition 2.15 and section 3 ) over XΣ . Here XΣ is the toric variety over S. Proof. We will explicitly construct a functor F : X(XΣ ,n) → X(Σ,Σ0n ) . To this aim, consider the skeleton πu : U → OXΣ for the identity element in Hom(XΣ , XΣ ), i.e., the tautological object (cf. [4, Theorem on page 10]). Observe that the log structure associated to πu : U → OXΣ is isomorphic to the canonical log structure MΣ . Indeed let us recall the construction of πu : U → OXΣ (cf. [4, page 11]). By [4, page 11], we have U = { union of subsheaves (σ ∨ ∩ M )Xσ of the constant sheaf M on XΣ of all σ ∈ Σ}. For a cone σ ∈ Σ, we have U(Xσ ) = σ ∨ ∩ M , and U(XΣ ) → OXΣ (Xσ ) is the natural map σ ∨ ∩ M → OS [σ ∨ ∩ M ]. Furthermore it is easy to see that if V ⊂ Xσ , then any element m ∈ U (V ) has the form m1 + m2 where m1 ∈ U(Xσ ) and m2 is an invertible element of U(V ). (If σ Â τ , then any element m ∈ τ ∨ ∩ M has the form m1 + m2 where m1 ∈ σ ∨ ∩ M and m2 is an invertible element of τ ∨ ∩ M .) Therefore πu : U → OXΣ induces a homomorphism ζ : U → MΣ which makes MΣ the log structure associated to U → MΣ . Let (f, h) : (X, α : M → OX ) → (XΣ , MΣ ) be an object in X(XΣ ,TΣ ,n) over f : X → XΣ . Define F ((f, h) : (X, α : M → OX ) → (XΣ , MΣ )) to be (f −1 πu : f −1 U → OX , α : M → OX , h ◦ (f −1 ζ) : f −1 U → M). Then (f −1 πu : f −1 U → OX , α : M → OX , h ◦ (f −1 ζ) : f −1 U → M) is an object in X(Σ,Σ0n ) because (f, h) is an admissible FR morphism of type n and the log structure MΣ is arising from the prelog structure πu : U → OXΣ . Clearly, a morphism of log structure φ : M → M commutes with h ◦ (f −1 ζ) : f −1 U → M if and only if it commutes with h : f ∗ MΣ → M because the log structure the f ∗ MΣ is arising from the natural morphism f −1 ζ

f −1 U → f −1 MΣ → f ∗ MΣ . Thus F is fully faithful over X → XΣ . Finally, we will show that F is essentially surjective over X → XΣ . Let (f −1 πu : f −1 U → OX , α : M → OX , l : f −1 U → M) be an object of X(Σ,Σ0n ) over f : X → XΣ . (Note that every object in X(Σ,Σ0n ) is of this form.) The homomorphism l : f −1 U → M induces the homomorphism la : f ∗ MΣ → M. Then F ((f, la ) : (X, M) → (XΣ , MΣ )) is isomorphic to (f −1 πu : f −1 U → OX , α : M → OX , l : f −1 U → M). Hence F is an isomorphism. 2

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25

Theorem 4.6. The stack X(Σ,Σ0n ) is a smooth tame algebraic stack that is locally of finite type over S and has finite diagonal. Furthermore it satisfies the additional properties such that: (1) The natural functor π(Σ,Σ0n ) : X(Σ,Σ0n ) −→ XΣ (cf. Definition 4.2 ) is a coarse moduli map, (2) The canonical torus embedding i(Σ,Σ0n ) : TΣ = Spec OS [M ] ,→ X(Σ,Σ0n ) is an open immersion identifying TΣ with a dense open substack of X(Σ,Σ0n ) . (3) If Σ is a finite fan, then X(Σ,Σ0n ) is of finite type over S. If (Σ, Σ0n ) is tame over the base scheme S, then X(Σ,Σ0n ) is a Deligne-Mumford stack over S. (Moreover there is a criterion for Deligne-Mumfordness. see Corollary 4.17.) If Σ is a non-singular fan and Σ0can denotes the canonical free net, then X(Σ,Σ0can ) is the toric variety XΣ over S. Proof. By Proposition 4.5, Proposition 3.5, and Proposition 3.6 (see also Remark 3.7), X(Σ,Σ0n ) is a smooth algebraic stack that is locally of finite type over S and has finite diagonal. Moreover π(Σ,Σ0n ) : X(Σ,Σ0n ) → XΣ is a coarse moduli map by Proposition 3.6 (2), and thus by Keel-Mori theorem (cf. [17], see also [7, Theorem 1.1]) π(Σ,Σ0n ) is proper. Hence if Σ is a finite fan, then X(Σ,Σ0n ) is of finite type over S. To see that X(Σ,Σ0n ) is tame, we may assume that the base scheme is a spectrum of a field because the tameness depends only on automorphism group schemes of geometric points (cf. [2, Theorem 3.2]). Thus the tameness follows from Proposition 4.5 and Corollary 3.10. Since the restriction MΣ |Spec OS [M ] of MΣ to TΣ = Spec OS [M ] is the trivial log structure, thus by taking −1 account of Proposition 4.5 the morphism π(Σ,Σ Hence 0 ) (TΣ ) → TΣ is an isomorphism. n 0 i(Σ,Σ0n ) : TΣ = Spec OS [M ] → X(Σ,Σ0n ) is an open immersion. If (Σ, Σn ) is tame over the base scheme S, the same argument as in Proof of algebraicity of Theorem 3.3 shows that X(Σ,Σ0n ) is a Deligne-Mumford stack. The last claim follows from Remark 4.4. 2 Definition 4.7. Let (Σ, Σ0n ) be a stacky fan. We shall call the stack X(Σ,Σ0n ) the toric algebraic stack associated to (Σ, Σ0n ) (or (Σ, n)). Remark 4.8. (1) Let (Smooth toric varieties over S), (resp. (Simplicial toric varieties over S)) denote the category of smooth (resp. simplicial) toric varieties over S whose morphisms are S-morphisms. Let (Toric algebraic stacks over S) denote the 2-category whose objects are toric algebraic stacks over S, and a 1-morphism is an S-morphism between objects, and a 2-morphism is an isomorphism between 1-morphisms. Given a 1-morphism f : X(Σ,Σ0n ) → X(∆,∆0 0 ) , by the universality of coarse moduli spaces, n there exists a unique morphism f0 : XΣ → X∆ such that π(∆,∆0 0 ) ◦ f = f0 ◦ π(Σ,Σ0n ) . n By attaching f0 to f , we obtain a functor c : (Toric algebraic stacks over S) → (Simplicial toric varieties over S), X(Σ,Σ0n ) 7→ XΣ . Therefore there is the following diagram of (2)-categories, a bbbbbbbbb1

(Smooth toric varieties over S) \ b \\\\\\-

(Toric algebraic stacks over S) c

²

(Simplicial toric varieties over S) where a and b is fully faithful functors and c is an essentially surjective functor.

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ISAMU IWANARI

(2) Gluing pieces together. Let (Σ, Σ0n ) be a stacky fan and σ be a cone in Σ. By definition there exists a natural fully faithful morphism X(σ,σn0 ) ,→ X(Σ,Σ0n ) where (σ, σn0 ) is the restriction of (Σ, Σ0n ) to σ. The image of this functor is identified with −1 the open substack π(Σ,Σ 0 ) (Xσ ), where Xσ (⊂ XΣ ) is the affine toric variety associated n to σ. That is to say, Definition 4.2 allows one to have a natural gluing construction ∪σ∈Σ X(σ,σn0 ) = X(Σ,Σ0n ) . Proposition 4.9. Let (σ, σn0 ) be a stacky fan such that σ is a simplicial cone in NR where N = Zd . Here N ∼ = Zd . Suppose that dim(σ) = r. Then the toric algebraic stack X(σ,σn0 ) has a finite fppf morphism p : ArS × Gd−r m,S → X(σ,σn0 ) where ArS is an r-dimensional affine space over S. Furthermore X(σ,σn0 ) is isomorphic to the quotient stack [ArS /G] × Gd−r where G is a finite flat group scheme over S. If σ is a m full-dimensional cone, the quotient [ArS /G] coincides with the quotient given in section 3.2 (cf. Proposition 3.5). If (σ, σn0 ) is tame over the base scheme S, then we can choose p to be a finite ´etale cover and G to be a finite ´etale group scheme. Proof. As in Remark 3.7 we choose a splitting N∼ = N 0 ⊕ N 00 (σ, σn0 ) ∼ = (τ, τn0 ) ⊕ {0} Xσ ∼ = Xτ × Gd−r m,S

where σ ∼ = τ ⊂ NR0 is a full-dimensional cone. Notice that P := τ ∨ ∩ M 0 is a simplicially toric sharp monoid. Let ι : P → F ∼ = Nr be the homomorphism of monoids defined as n the composite P → F → F where P → F is the minimal free resolution and n : F → F is defined by eρ 7→ nρ · eρ for each ray ρ ∈ σ(1). Here eρ is the irreducible element of F corresponding to ρ (cf. Lemma 2.14). Then by Proposition 3.5 and Proposition 4.5, the stack X(τ,τn0 ) is isomorphic to the quotient stack [Spec OS [F ]/G] where G is the Cartier dual (F gp /ι(P )gp )D . The group scheme G is a finite flat over S. We remark that X(σ,σn0 ) ∼ = d−r X(τ,τn0 ) × Gm,S by Proposition 4.5, and thus our claim follows. The last assertion is clear because in such case G is a finite ´etale group scheme. 2 Proposition 4.10. The toric algebraic stack X(Σ,Σ0n ) is proper over S if and only if (Σ, Σ0n ) is complete (cf. section 4.1). Proof. If Σ is a finite and complete fan, then since the coarse moduli map π(Σ,Σ0n ) is proper, properness of X(Σ,Σ0n ) over S follows from the fact that XΣ is proper over S (cf. [10, Proposition in page 39] or [6, Chapter IV, Theorem 2.5 (viii)]). Conversely, suppose that X(Σ,Σ0n ) is proper over S. We will show that Σ is a finite fan and the support |Σ| is the whole space NR . By [23, Lemma 2.7(ii)], we see that XΣ is proper over S. By ([10, Proposition in page 39] or [6, Chapter IV, Theorem 2.5 (viii)]), it suffices to show only that Σ is a finite fan (this is well known, but we give the proof here because [10] assumes that all fans are finite). Let Σmax be the subset of Σ consisting of the maximal elements with respect to the face relation. The set {Xσ }σ∈Σmax is an open covering of XΣ , but {Xσ }σ∈Σ0 is not an open covering for any proper subset Σ0 ⊂ Σmax because for any cone σ,

LOG GEOMETRY AND TORIC ALGEBRAIC STACKS

27

set-theoretically Xσ = tσÂτ Zτ (cf. section 1.1). Since XΣ is quasi-compact, Σmax is finite. Hence Σ is a finite fan (note that Σ is simplicial in our case). 2 Remark 4.11. Here we will explain the relationship between toric Deligne-Mumford stacks introduced in [5] and toric algebraic stacks introduced in this section. We assume that the base scheme is an algebraically closed field of characteristic zero (since the results of [5] only hold in characteristic zero). First, since both took different approaches, it is not clear that if one begins with a given stacky fan, then two associated stacks in the sense of [5] and us are isomorphic to each other. But in the subsequent paper [12] we prove the geometric characterization theorem for toric algebraic stacks in the sense of us. One can deduce from it that they are (non-canonically) isomorphic to each other. (See [12, Section 5].) Next, toric Deligne-Mumford stacks in the sense of [5] admits a finite abelian gerbe structure, while our toric algebraic stacks do not. However, such structures can be obtained form our toric algebraic stacks by the following well-known technique. We here work over Z. Let LD be an invertible sheaf on X(Σ,Σ0n ) , associated to a torus invariant divisor D (see the next subsection). Consider the triple (u : U → X(Σ,Σ0n ) , M, φ : u∗ LD ∼ = M⊗n ) where M is an invertible sheaf on U , and φ is an isomorphism of sheaves. Morphisms of 1/n triples are defined in the natural manner. Then it forms an algebraic stack X(Σ,Σ0n ) (LD ) 1/n and there exists the natural forgetting functor X(Σ,Σ0n ) (LD ) → X(Σ,Σ0n ) , which is a smooth morphism. The composition of this procedure, that is, 1/n

1/n

X(Σ,Σ0n ) (LD1 1 ) ×X(Σ,Σ0 ) · · · ×X(Σ,Σ0 ) X(Σ,Σ0n ) (LDr r ) → X(Σ,Σ0n ) n

n

1/n

yields a gerbe that appears on toric Deligne-Mumford stacks. The stack X(Σ,Σ0n ) (LD ) associated to LD can be viewed as the fiber product 1/n

X(Σ,Σ0n ) (LD )

/ BG m

²

² / BGm ,

X(Σ,Σ0n )

where the lower horizontal arrow is associated to LD , BGm is the classifying stack of Gm , and BGm → BGm is associated to the homomorphism Gm → Gm , a 7→ an . This interpretation has the following direct generalization. Let N be a finitely generated abelian group and h : Z → N be a homomorphism of abelian groups. Then it gives rise to G = Spec Z[N ] → Gm and BG → BGm . Then we can define X(Σ,Σ0n ) (LD , h) to be the fiber product BG X(Σ,Σ0n )

² / BG . m

Notice that X(Σ,Σ0n ) (LD , h) is smooth, but does not have the finite inertia stack, that is, automorphism groups of objects in X(Σ,Σ0n ) (LD , h) can be positive dimensional.

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4.3. Torus-invariant cycles and log structures on toric algebraic stacks. In this subsection we define torus-invariant cycles and a canonical log structure on toric algebraic stacks. It turns out that if a base scheme S is regular the complement of a torus embedding in a toric algebraic stack is a divisor with normal crossings relative to S (unlike simplicial toric varieties), and it fits nicely in the notions of torus-invariant divisors and canonical log structure. Torus-invariant cycles. Let (Σ ⊂ NR , Σ0n ) be a stacky fan, where N = Zd . The torusinvariant cycle V (σ) ⊂ XΣ associated to σ is represented by the functor FV (σ) : (S-schemes) → (Sets), X 7→ {all pairs (S, π : S → OX ) of skeletons with property (∗)} where (∗) = {For any point x ∈ X, there exists a cone τ such that σ ≺ τ and Sx = τ ∨ ∩ M, and π(s) = 0 if s ∈ τ ∨ ∩ σo∨ ∩ M ⊂ Sx } (see section 1.1 for the definition of σo∨ ). Indeed, to see this, we may and will suppose that XΣ is an affine toric variety Xτ = Spec OS [τ ∨ ∩ M ] with σ ≺ τ . Let π : U → OXτ be the skeleton for the identity morphism Xτ → Xτ . As in the proof of Proposition 4.5, U = { union of subsheaves (γ ∨ ∩ M )Xγ of the constant sheaf M on Xτ of all γ ≺ τ }. Let U 0 := { union of subsheaves (γ ∨ ∩ σo∨ ∩ M )Xγ of the constant sheaf M on Xτ of all γ ≺ τ }. Then FV (σ) is represented by Spec(OS [τ ∨ ∩ M ]/π(U 0 )). Since γ ∨ ∩ σo∨ ∩ M = U 0 (Xγ ) ⊂ γ ∨ ∩ M = U(Xγ ) for γ ≺ τ , it is easy to check that Spec(OS [τ ∨ ∩ M ]/π(U 0 )) = Spec(OS [τ ∨ ∩ M ]/τ ∨ ∩ σo∨ ∩ M ) = V (σ). Now let us define the torus-invariant cycle V (σ) associated to σ ∈ Σ. Consider the substack V (σ) of X(Σ,Σ0n ) , that consists of objects (π : S → OX , α : M → OX , η : S → M) such that the condition (∗∗) holds, where (∗∗) = { For any geometric point x¯ → X, α(m) = 0 if m ∈ Mx¯ and there exists a positive integer n such that the image of n · m in Mx¯ lies in η(σo∨ ∩ Sx¯ )}. Note that if Σ is non-singular and Σ0n is the canonical free net, i.e., X(Σ,Σ0n ) = XΣ , then (∗∗) is equal to (∗) for any cone in Σ. Clearly, the substack V (σ) is stable under the torus action. The following lemma shows that V (σ) is a closed substack of X(Σ,Σ0n ) . Lemma 4.12. With the same notation as above, the condition (∗∗) is represented by a closed subscheme of X. In particular, V (σ) ⊂ X(Σ,Σ0n ) is a closed substack. Moreover if the base scheme S is reduced, then V (σ) is reduced. Proof. Let φ : X → X(Σ,Σ0n ) be the morphism corresponding to (π : S → OX , α : M → OX , η : S → M). Since our claim is smooth local on X, thus we may assume that Σ is a full-dimensional simplicial cone τ and σ ≺ τ . Then by Proposition 4.9, we have the diagram p

π(τ,τ 0 )

n Spec OS [F ] −−−→ X(τ,τn0 ) −−−→ Xτ = Spec OS [τ ∨ ∩ M ]

i

n

where p is an fppf morphism and the composite is defined by τ ∨ ∩ M ,→ F ,→ F . Here i is the minimal free resolution and n is defined by eρ 7→ nρ · eρ , where eρ is the irreducible element of F corresponding to ρ ∈ τ (1). Since p is fppf, there exists ´etale locally a lifting X → Spec OS [F ] of φ : X → X(Σ,Σ0n ) , and thus we can assume X := Spec OS [F ] and that the log structure M is induced by the natural map F → OS [F ] (cf. Proposition 2.9). What we have to prove is that the condition (∗∗) is represented by a closed subscheme on

LOG GEOMETRY AND TORIC ALGEBRAIC STACKS

29

X. Consider the following natural diagram / τ ∨ ∩ M n◦i / F NNN xx NNN xx x NNN xx NN' ² |xx h

τ ∨ ∩ σ0∨ ∩ M

OS [F ] Set A = {f ∈ F | there is n ∈ Z≥1 such that n · f ∈ Image(τ ∨ ∩ σ0∨ ∩ M )}. Let I be the ideal of OS [F ] that is generated by h(A). We claim that the closed subscheme Spec OX /I represents the condition (∗∗). To see this, note first that τ ∨ ∩ σ0∨ ∩ M is (τ ∨ ∩ M ) r (σ ⊥ ∩ τ ∨ ∩ M ) and by embeddings (τ ∨ ∩ M ) ⊂ F ⊂ τ ∨ , each irreducible element of F lies on a unique ray of τ ∨ (cf. Proposition 2.9 (2)). Set F = Nr ⊕ Nd−r and suppose that Nd−r generates the face (σ ⊥ ∩ τ ∨ ) of τ ∨ . Since τ ∨ ∩ M is close to F , thus I is the ideal generated by Nr ⊕ Nd−r r {0} ⊕ Nd−r (⊂ F ). Let ei be the i-th standard irreducible element of F = Nd . Let x¯ → X = Spec OS [F ] be a geometric point such that Sx¯ = γ ∨ ∩ M where σ ⊂ γ ⊂ τ , and let Mx¯ = hei1 , . . . , eis i ⊂ F. In order to prove our claim, let us show that in OX,¯x , the ideal I coincides with the ideal generated by L := {m ∈ Mx¯ | there is n ∈ Z≥1 such that n · m ∈ Image(γ ∨ ∩ σ0∨ ∩ M )}. After reordering, we suppose 1 ≤ i1 <, . . . , < it ≤ r < it+1 <, . . . , ≤ is (t ≤ s). Then we have (OX /I)x¯ = OX,¯x /(ei1 , . . . , eit ). Consider the following natural commutative diagram γ ∨ ∩ σ0∨ ∩ M

/ γ∨ ∩ M ²

/ (γ ∨ ∩ M )gp

/ M ⊗Z Q

/ F gp ⊗Z Q

/M x ¯

/M x ¯

² / Mgp ⊗ Q. Z x ¯

=

Sx¯

Since τ ∨ ∩ σ0∨ ∩ M is equal to (τ ∨ ∩ M ) r (σ ⊥ ∩ τ ∨ ∩ M ) and the image of σ ⊥ (⊂ M ⊗Z Q) gp in Mx¯ ⊗Z Q is the vector space spanned by eit+1 , . . . , eis , thus the image of γ ∨ ∩ σ0∨ ∩ M in Mx¯ is contained in the set L0 := N · ei1 ⊕ · · · ⊕ N · eis r N · eit+1 ⊕ · · · ⊕ N · eis in Mx¯ . Since the image of γ ∨ ∩ M in Mx¯ is close to Mx¯ , we have L = L0 . Therefore the ideal of OX,¯x generated by ei1 , . . . , eit is the ideal of OX,¯x generated by L. Hence the ideal I coincides with the ideal generated by L in OX,¯x and thus they are the same in OX because I is finitely generated. Thus Spec OX /I represents the condition (∗∗). Finally, we will show the last assertion. Suppose that S is reduced. With the same notation and assumption as above, OS [F ]/I is reduced. Since Spec OS [F ] → X(τ,τn0 ) is fppf, thus V (σ) is reduced. 2 Definition 4.13. We shall call the closed substack V (σ) the torus-invariant cycle associated to σ. Proposition 4.14. Let τ and σ be cones in a stacky fan (Σ, Σ0n ) and suppose that σ ≺ τ . Then we have V (τ ) ⊂ V (σ), i.e., V (τ ) is a closed substack of V (σ). Proof. By the definition and Proposition 4.12, it is enough to prove that for any cone γ such that σ ≺ τ ≺ γ, we have γ ∨ ∩ σ0∨ ∩ M ⊂ γ ∨ ∩ τ0∨ ∩ M . Clearly γ ∨ ∩ σ0∨ ⊂ γ ∨ ∩ τ0∨ , thus our claim follows. 2 Stabilizer groups. By the calculation of stabilizer group scheme in section 3.4 and Proposition 4.5, we can easily calculate the stabilizer group schemes of points on a toric algebraic stack.

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Proposition 4.15. Let (Σ, Σ0n ) be a stacky fan and X(Σ,Σ0n ) the associated stack. Let x¯ : Spec K → X(Σ,Σ0n ) be a geometric point on X(Σ,Σ0n ) such that x¯ lies in V (σ), but does not lie in any torus-invariant proper substack V (τ ) (σ ≺ τ ). Here K is an algebraically closed S-field. Let S → OK be the skeleton for π(Σ,Σ0n ) ◦ x¯ : Spec K → X(Σ,Σ0n ) → XΣ . Then Sx¯ = σ ∨ ∩ M , and the stabilizer group scheme G at x¯ is the Cartier dual of F gp /ι((P )gp . Here P := (σ ∨ ∩ M )/(invertible elements) and ι : P → F is t ◦ r in Definition 4.2 (6). The rank of the stabilizer group scheme over x¯, i.e., dimK Γ(G, OG ) is mult(σ, σn0 ) = mult(σ) · Πρ∈σ(1) nρ . Proof. By our assumption and the definition of V (σ), clearly, Sx¯ = σ ∨ ∩ M . Taking account of Proposition 3.9 and Proposition 4.5, the stabilizer group scheme G at x¯ is the Cartier dual of F gp /ι((P )gp . To see the last assertion, it suffices to show that the order of F gp /ι((P )gp is equal to mult(σ) · Πρ∈σ(1) nρ . It follows from the fact that the multiplicity of P is equal to mult(σ) (cf. Remark 2.7 (3)). 2 Remark 4.16. The order of stabilizer group on a point on a Deligne-Mumford stack is an important invariant to intersection theory with rational coefficients on it (cf. [26]). The stabilizer groups on a toric algebraic stacks are fundamental invariants because they have data arising from levels of the stacky fan and multiplicities of cones in the stacky fan. In particular, the stabilizer group scheme on a generic point on a torus-invariant divisor V (ρ) (ρ ∈ Σ(1)) is isomorphic to µnρ . Corollary 4.17. Let (Σ, Σ0n ) be a stacky fan and X(Σ,Σ0n ) the associated stack. Then X(Σ,Σ0n ) is a Deligne-Mumford stack if and only if (Σ, Σ0n ) is tame over S. Proof. First of all, the “if” direction follows from Theorem 4.6. Thus we will show the “only if” direction. Assume that (Σ, Σ0n ) is not tame over S. Then there exists a cone σ ∈ Σ such that r := mult(σ, σn0 ) is not invertible on S. Then there exists an algebraically closed S-field K such that the characteristic of K is a prime divisor of r. Consider a point w : Spec K → X(Σ,Σ0n ) ×S K that corresponds to a triple (S → OSpec K , M → OSpec K , S → M) such that S = σ ∨ ∩ M . Then by Proposition 4.15, the stabilizer group scheme of w is the Cartier dual of a finite group that has the form Z/pc11 Z ⊕ · · · ⊕ Z/pcl l Z such that pi ’s are prime numbers and r = Π0≤i≤l pci i . After reordering, assume that p1 is the characteristic of K. Then the stabilizer group scheme of w is not ´etale over K. Thus by [19, Lemma 4.2], X(Σ,Σ0n ) is not a Deligne-Mumford stack. Hence our claim follows. 2 Remark 4.18. The first version [11] of this paper only treats the case when toric algebraic stacks are Deligne-Mumford stacks, although our method can apply to the case over arbitrary base schemes. Log structure on X(Σ,Σ0n ) . A canonical log structure on Lis-´et(X(Σ,Σ0n ) ) is defined as follows. To a smooth morphism X → X(Σ,Σ0n ) corresponding to a triple (π : S → OX , α : M → OX , η : S → M), we attach the log structure α : M → OX . It gives rise to a fine and saturated log structure M(Σ,Σ0n ) on Lis-´et(X(Σ,Σ0n ) ). We shall refer it as the canonical log structure. Through the identification X(Σ,Σ0n ) = X(Σ,TΣ ,n) (cf. Proposition 4.5), it is equivalent to the canonical log structure M(XΣ ,TΣ ,n) on X(Σ,TΣ ,n) . As in section 3.5, we have a natural morphism of log stacks (π(Σ,Σ0n ) , φ(Σ,Σ0n ) ) : (X(Σ,Σ0n ) , M(Σ,Σ0n ) ) → (XΣ , MΣ ) whose underlying morphism is the canonical coarse moduli map.

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Theorem 4.19. Assume that the base scheme S is regular. Then the complement D = X(Σ,Σ0n ) − TΣ with reduced closed substack structure is a divisor with normal crossings relative to S, and the log structure M(Σ,Σ0n ) is arising from D. Furthermore D coincides with the union of ∪ρ∈Σ(1) V (ρ). For the proof we need the following lemma. Lemma 4.20 (log EGA IV 17.7.7). Let (f, φ) : (X, L) → (Y, M) and (g, ψ) : (Y, M) → (Z, N ) be morphisms of fine log schemes respectively. Suppose that (g, ψ) ◦ (f, φ) is log smooth and g is locally of finite presentation. Assume further that (f, φ) is a strict faithfully flat morphism that is locally of finite presentation. Then (g, ψ) is log smooth. Proof. Let (T0 , P0 ) ²

s

i

(T, P)

t

/ (Y, M) ² / (Z, N )

be a commutative diagram of fine log schemes where i is a strict closed immersion defined by a nilpotent ideal I ⊂ OT . It suffices to show that there exists ´etale locally on T a morphism (T, P) → (Y, M) filling in the diagram. To see this, after replacing T0 by an ´etale cover we may suppose that s has a lifting s0 : (T0 , P0 ) → (X, L). Since (g, ψ) ◦ (f, φ) is log smooth, there exists ´etale locally on T a lifting (T, P) → (X, L) of s0 filling in the diagram. Hence our assertion easily follows. 2 Proof of Theorem 4.19. By Proposition 4.5 and Proposition 3.5 (together with Remark 3.7), there is an fppf strict morphism (X, L) → (X(Σ,Σ0n ) , M(Σ,Σ0n ) ) from a scheme X such that the composite (X, L) → (S, OS∗ ) is log smooth. Here OS∗ denotes the trivial log structure on S. Then by applying Lemma 4.20 we see that (X(Σ,Σ0n ) , M(Σ,Σ0n ) ) → (S, OS∗ ) is log smooth. Let U → X(Σ,Σ0n ) be a smooth surjective morphism from an S-scheme U . Then by [14, Theorem 4.8] there exists ´etale locally on U a smooth morphism U → S ×Z Z[P ] where P is a toric monoid ([14, Theorem 4.8] worked over a field, but it is also applicable to our case). In addition, the natural map P ,→ OS [P ] gives rise to a chart for M(Σ,Σ0n ) . Since X(Σ,Σ0n ) is smooth over S, thus after shrinking S ×Z Z[P ] we may suppose that P is a free monoid. Note that the support of M(Σ,Σ0n ) is the complement X(Σ,Σ0n ) − TΣ . Therefore X(Σ,Σ0n ) − TΣ with reduced substack structure is a divisor with normal crossings relative to S. Next we will prove that ∪ρ∈Σ(1) V (ρ) is reduced. (Settheoretically ∪ρ∈Σ(1) V (ρ) = X(Σ,Σ0n ) − TΣ .) It follows from Lemma 4.12. Finally, we will show M(Σ,Σ0n ) ∼ = MD where MD is the log structure associated to D. Note that MD = OX(Σ,Σ0 ) ∩ i(Σ,Σ0n )∗ OT∗Σ where i(Σ,Σ0n ) : TΣ → X(Σ,Σ0n ) is the canonical torus emn bedding. Hence MD is a subsheaf of OX(Σ,Σ0 ) . Thus we may and will assume that n X(Σ,Σ0n ) = X(τ,τn0 ) where τ is a full-dimensional cone. Let p : Spec OS [F ] → X(τ,τn0 ) be an fppf cover given in Proposition 4.9. By the fppf descent theory for fine log structures ([22, Corollary A.5]) together with the fact that MD is a subsheaf OX(τ,τ 0 ) , it is enough to n show p∗ M(τ,τn0 ) ∼ = p∗ MD . Since p∗ M(τ,τn0 ) is induced by the natural map F → OS [F ] (cf. Proposition 4.5 and Proposition 2.16), thus we have p∗ M(τ,τn0 ) ∼ 2 = p∗ MD .

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Proposition 4.21. With the same notation and assumptions as above, suppose further that (Σ, Σ0n ) is tame over S. Then (π(Σ,Σ0n ) , φ(Σ,Σ0n ) ) : (X(Σ,Σ0n ) , M(Σ,Σ0n ) ) → (XΣ , MΣ ) is Kummer log ´etale. In particular, there exists an isomorphism of OX(Σ,Σ0 ) -modules n

∼

Ω1 ((X(Σ,Σ0n ) , M(Σ,Σ0n ) ))/(S, OS∗ )) → OX(Σ,Σ0 ) ⊗Z M, n

1

where Ω

((X(Σ,Σ0n ) , M(Σ,Σ0n ) ))/(S, OS∗ ))

is the sheaf of log differentials (cf. [14, 5.6]).

Proof. It follows from the next Lemma, Proposition 4.5 and Proposition 3.5.

2

Lemma 4.22. With the same notation as in Proposition 2.16, if the order of F gp /ιgp (P gp ) is invertible on R, then (f, φ) : (Spec R[F ], MF ) → (Spec R[P ], MP ) is Kummer log ´etale. In particular, the induced morphism (cf. section 3.2) ([Spec R[F ]/G], M) → (Spec R[P ], MP ) is Kummer log ´etale. Here G is the Cartier dual of F gp /ι(P gp ) and M is the log structure associated to a natural chart F → R[F ]. Proof. Since (f, φ) is an admissible FR morphism, thus (f, φ) is Kummer. By the toroidal characterization of log ´etaleness ([15, Theorem 3.5]), (f, φ) : (Spec R[F ], MF ) → (Spec R[P ], MP ) is log ´etale. Since Spec R[F ] → [Spec R[F ]/G] is ´etale, thus our claim follows. 2 Proof of Proposition 3.11. With the same notation as in Proposition 3.11, we may assume that X is an affine simplicial toric variety. Then Proposition 3.11 follows from Proposition 4.5, Theorem 4.19 and Proposition 4.21. 2 References [1] D. Abramovich, Lectures on Gromov-Witten invariants of orbifolds, arXiv:math.AG/0512372. [2] D. Abramovich, M. Olsson, and A. Vistoli, Tame stacks in positive characteristic, preprint (2007). [3] D. Abramovich and A. Vistoli, Compactifying the space of stable maps, Jour. Amer. Math. Soc. 15 (2002) no.1 p27–75. [4] A. Ash, D. Mumford, M. Rapoport, and Y.-S. Tai, Smooth compactifications of locally symmetric varieties, Math. Sci. Press, Bookline, MA, (1975) [5] L. Borisov, L. Chen, and G. Smith, The orbifold Chow rings of toric Deligne-Mumford stacks, Jour. Amer. Math. Soc. 18 (2005) 193–215. [6] C.-L. Chai and G. Faltings, Degeneration of abelian varieties, Springer-Verlag (1990). [7] B. Conrad, The Keel-Mori theorem via stacks, preprint available on his homepage. [8] D. Cox, The homogeneous coordinate ring of a toric variety, J. Alg. Geom. 4 (1995) 17–50. ´ ements de g´eom´etrie alg´ebrique, Inst. Hautes Etudes ´ [9] J. Dieudonn´e and A. Grothendieck, El´ Sci. Publ. Math. 4, 8, 11, 17, 20, 24, 28, 32 (1961–1967). [10] W. Fulton, Introduction to Toric Varieties, Ann. Math. Stud. Princeton Univ. Press (1993). [11] I. Iwanari, Toroidal geometry and Deligne-Mumford stacks, preprint series of Kyoto univ., July (2006). [12] I. Iwanari, The category of toric stacks, Compositio Math. 145 (2009) 718–746. [13] I. Iwanari, Integral Chow rings of toric stacks, Internat. Math. Res. Notices (2009) rnp110. [14] F. Kato, Log smooth deformation theory, Tohoku Math. J. 48 (1996) 317–354. [15] K. Kato, Logarithmic structure of Fontaine-Illusie, Algebraic analysis, geometry and number theory (Baltimore, MD, 1988), 191–224 Johns Hopkins Univ. Press, Baltimore, MD, 1989. [16] K. Kato, Toric singularities, Amer. J. Math. 116 (1994), 1073–1099. [17] S. Keel and S. Mori, Quotients by groupoids, Ann. Math. 145 (1997), 193–213.

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[18] G. Kempf, F. Knudsen, D. Mumford, B. Saint-Donat, Toroidal Embeddings 1, Lecture Note in Math. 339 Springer-Verlag (1973). [19] G. Laumon and L. Moret-Bailly, Champs Alg´ebriques, Springer-Verlag (2000). [20] Y. Namikawa, Toroidal compactification of Siegel spaces, Lecture Note in Math. 812 (1980) SpringerVerlag. [21] T. Oda, Convex bodies and algebraic geometry, Springer-Verlag, 1988. ´ [22] M. Olsson, Logarithmic geometry and algebraic stacks, Ann. Sci. Ecole Norm. Sup. 36 (2003), 747– 791. [23] M. Olsson, Hom-stacks and restriction of scalars, Duke Math. J. 134 (2006), 139–164. [24] M. Romagny, Group actions on stacks and applications, Michigan Math. J. 53 (2005), no. 1, 209–236. [25] H. Thompson, Toric singularities revisited, preprint (2005) arXiv:math.AC/0305441 [26] A. Vistoli, Intersection theory on algebraic stacks and their moduli spaces, Invent. Math. 97 (1989), 613–670.